S3: Phase planes

Example 1

S3: Phase planes (ver. 1)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x + y + 15\]

\[y'= -x - 5 \, y + 9\]

Answer.

The isoclines cross at \(( 4 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 2

S3: Phase planes (ver. 2)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 5 \, y - 25\]

\[y'= -2 \, x - 2 \, y + 8\]

Answer.

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 3

S3: Phase planes (ver. 3)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 4 \, y + 19\]

\[y'= 4 \, x + 4 \, y - 12\]

Answer.

The isoclines cross at \(( -1 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 4

S3: Phase planes (ver. 4)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x + 3 \, y + 7\]

\[y'= -3 \, x - 5 \, y - 7\]

Answer.

The isoclines cross at \(( 1 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 5

S3: Phase planes (ver. 5)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + 3 \, y - 10\]

\[y'= -2 \, x - 3 \, y - 10\]

Answer.

The isoclines cross at \(( -5 , 0 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 6

S3: Phase planes (ver. 6)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 4 \, y + 4\]

\[y'= -2 \, x - 4 \, y + 24\]

Answer.

The isoclines cross at \(( 4 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 7

S3: Phase planes (ver. 7)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - 4 \, y - 8\]

\[y'= -3 \, x - 4 \, y + 24\]

Answer.

The isoclines cross at \(( 4 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 8

S3: Phase planes (ver. 8)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 3 \, y + 35\]

\[y'= x - 5 \, y + 21\]

Answer.

The isoclines cross at \(( 4 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 9

S3: Phase planes (ver. 9)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 2 \, y - 6\]

\[y'= -3 \, x - 3 \, y - 3\]

Answer.

The isoclines cross at \(( 1 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 10

S3: Phase planes (ver. 10)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 5 \, y\]

\[y'= -x + 2 \, y + 7\]

Answer.

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 11

S3: Phase planes (ver. 11)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x + 2 \, y\]

\[y'= -x - 2 \, y\]

Answer.

The isoclines cross at \(( 0 , 0 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 12

S3: Phase planes (ver. 12)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 3 \, y + 3\]

\[y'= 5 \, x - 2 \, y + 2\]

Answer.

The isoclines cross at \(( 0 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 13

S3: Phase planes (ver. 13)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - y - 8\]

\[y'= -2 \, x + y - 10\]

Answer.

The isoclines cross at \(( -3 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 14

S3: Phase planes (ver. 14)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 3 \, y - 2\]

\[y'= -3 \, x - 2 \, y - 23\]

Answer.

The isoclines cross at \(( -5 , -4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 15

S3: Phase planes (ver. 15)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - 4 \, y + 16\]

\[y'= -x + 4 \, y - 6\]

Answer.

The isoclines cross at \(( 2 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 16

S3: Phase planes (ver. 16)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - y - 3\]

\[y'= -5 \, x + y - 6\]

Answer.

The isoclines cross at \(( -1 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 17

S3: Phase planes (ver. 17)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 2 \, y + 2\]

\[y'= -5 \, x - 3 \, y - 8\]

Answer.

The isoclines cross at \(( -1 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 18

S3: Phase planes (ver. 18)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 4 \, y - 11\]

\[y'= 2 \, x - 5 \, y - 15\]

Answer.

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 19

S3: Phase planes (ver. 19)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x + 4 \, y - 18\]

\[y'= 3 \, x - 2 \, y + 13\]

Answer.

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 20

S3: Phase planes (ver. 20)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 5 \, y + 14\]

\[y'= -x - 2 \, y + 3\]

Answer.

The isoclines cross at \(( -1 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 21

S3: Phase planes (ver. 21)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + 4 \, y + 14\]

\[y'= -4 \, x - 5 \, y - 37\]

Answer.

The isoclines cross at \(( -3 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 22

S3: Phase planes (ver. 22)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 5 \, y - 3\]

\[y'= 5 \, x - 3 \, y + 15\]

Answer.

The isoclines cross at \(( -3 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 23

S3: Phase planes (ver. 23)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x - 5 \, y - 28\]

\[y'= -x + 3 \, y + 8\]

Answer.

The isoclines cross at \(( -4 , -4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 24

S3: Phase planes (ver. 24)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + 2 \, y - 10\]

\[y'= -3 \, x - 5 \, y + 17\]

Answer.

The isoclines cross at \(( -1 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 25

S3: Phase planes (ver. 25)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x + 2 \, y - 6\]

\[y'= 4 \, x - 2 \, y + 6\]

Answer.

The isoclines cross at \(( 0 , 3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 26

S3: Phase planes (ver. 26)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - 5 \, y - 5\]

\[y'= -4 \, x - 3 \, y + 25\]

Answer.

The isoclines cross at \(( 4 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 27

S3: Phase planes (ver. 27)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - y + 11\]

\[y'= 4 \, x + y + 13\]

Answer.

The isoclines cross at \(( -4 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 28

S3: Phase planes (ver. 28)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - 4 \, y + 16\]

\[y'= 2 \, x - 2 \, y + 12\]

Answer.

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 29

S3: Phase planes (ver. 29)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 2 \, y - 7\]

\[y'= -x - 3 \, y + 6\]

Answer.

The isoclines cross at \(( 3 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 30

S3: Phase planes (ver. 30)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - 4 \, y + 4\]

\[y'= -x + 4 \, y - 14\]

Answer.

The isoclines cross at \(( -2 , 3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 31

S3: Phase planes (ver. 31)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 3 \, y + 9\]

\[y'= x - 5 \, y + 15\]

Answer.

The isoclines cross at \(( -5 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 32

S3: Phase planes (ver. 32)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 4 \, y + 4\]

\[y'= 5 \, x - 2 \, y + 2\]

Answer.

The isoclines cross at \(( 0 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 33

S3: Phase planes (ver. 33)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - y + 19\]

\[y'= -4 \, x - 4 \, y - 24\]

Answer.

The isoclines cross at \(( -5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 34

S3: Phase planes (ver. 34)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x - 5 \, y + 22\]

\[y'= 5 \, x - 3 \, y + 7\]

Answer.

The isoclines cross at \(( 1 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 35

S3: Phase planes (ver. 35)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - 4 \, y + 31\]

\[y'= -5 \, x - 4 \, y + 1\]

Answer.

The isoclines cross at \(( -3 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 36

S3: Phase planes (ver. 36)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 3 \, y + 1\]

\[y'= 2 \, x - 2 \, y - 10\]

Answer.

The isoclines cross at \(( 4 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 37

S3: Phase planes (ver. 37)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -3 \, x - y + 11\]

\[y'= 2 \, x - 2 \, y - 2\]

Answer.

The isoclines cross at \(( 3 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 38

S3: Phase planes (ver. 38)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x + y - 12\]

\[y'= -4 \, x - y - 15\]

Answer.

The isoclines cross at \(( -3 , -3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 39

S3: Phase planes (ver. 39)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x + 5 \, y + 9\]

\[y'= 2 \, x - y - 9\]

Answer.

The isoclines cross at \(( 3 , -3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 40

S3: Phase planes (ver. 40)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 3 \, y - 7\]

\[y'= -4 \, x + y - 2\]

Answer.

The isoclines cross at \(( -1 , -2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 41

S3: Phase planes (ver. 41)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 3 \, y + 19\]

\[y'= -5 \, x - 4 \, y + 10\]

Answer.

The isoclines cross at \(( -2 , 5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 42

S3: Phase planes (ver. 42)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x + y - 2\]

\[y'= 4 \, x - 2 \, y + 12\]

Answer.

The isoclines cross at \(( -1 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 43

S3: Phase planes (ver. 43)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - 5 \, y + 25\]

\[y'= -4 \, x - 3 \, y - 6\]

Answer.

The isoclines cross at \(( -3 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 44

S3: Phase planes (ver. 44)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + 4 \, y + 10\]

\[y'= -x - 3 \, y - 20\]

Answer.

The isoclines cross at \(( -5 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 45

S3: Phase planes (ver. 45)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - y + 17\]

\[y'= 2 \, x - 2 \, y - 6\]

Answer.

The isoclines cross at \(( 4 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 46

S3: Phase planes (ver. 46)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - y + 5\]

\[y'= -2 \, x - 5 \, y + 11\]

Answer.

The isoclines cross at \(( -2 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 47

S3: Phase planes (ver. 47)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - y - 10\]

\[y'= -2 \, x - y + 6\]

Answer.

The isoclines cross at \(( 4 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 48

S3: Phase planes (ver. 48)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x + y + 16\]

\[y'= 5 \, x - 4 \, y + 20\]

Answer.

The isoclines cross at \(( -4 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 49

S3: Phase planes (ver. 49)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -3 \, x - y - 3\]

\[y'= -5 \, x + 4 \, y + 12\]

Answer.

The isoclines cross at \(( 0 , -3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 50

S3: Phase planes (ver. 50)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 3 \, y - 19\]

\[y'= -2 \, x - 2 \, y - 8\]

Answer.

The isoclines cross at \(( 1 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 51

S3: Phase planes (ver. 51)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 4 \, y + 1\]

\[y'= 5 \, x + y + 17\]

Answer.

The isoclines cross at \(( -3 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 52

S3: Phase planes (ver. 52)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 3 \, y - 6\]

\[y'= 5 \, x - 3 \, y + 22\]

Answer.

The isoclines cross at \(( -2 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 53

S3: Phase planes (ver. 53)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 4 \, y + 1\]

\[y'= -x - 4 \, y + 7\]

Answer.

The isoclines cross at \(( 3 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 54

S3: Phase planes (ver. 54)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 5 \, y + 2\]

\[y'= -x - 3 \, y + 8\]

Answer.

The isoclines cross at \(( 2 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 55

S3: Phase planes (ver. 55)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x - 4 \, y + 26\]

\[y'= -3 \, x + 5 \, y - 5\]

Answer.

The isoclines cross at \(( 5 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 56

S3: Phase planes (ver. 56)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x + y - 1\]

\[y'= 3 \, x - 2 \, y - 9\]

Answer.

The isoclines cross at \(( 1 , -3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 57

S3: Phase planes (ver. 57)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 4 \, y - 20\]

\[y'= -4 \, x - 3 \, y - 1\]

Answer.

The isoclines cross at \(( 2 , -3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 58

S3: Phase planes (ver. 58)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 2 \, y\]

\[y'= -x + y + 3\]

Answer.

The isoclines cross at \(( 2 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 59

S3: Phase planes (ver. 59)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 5 \, y - 19\]

\[y'= 2 \, x + y + 11\]

Answer.

The isoclines cross at \(( -3 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 60

S3: Phase planes (ver. 60)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x + 3 \, y - 9\]

\[y'= -2 \, x - 3 \, y - 12\]

Answer.

The isoclines cross at \(( -3 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 61

S3: Phase planes (ver. 61)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 4 \, y\]

\[y'= 3 \, x - 2 \, y + 22\]

Answer.

The isoclines cross at \(( -4 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 62

S3: Phase planes (ver. 62)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 2 \, y + 8\]

\[y'= -3 \, x - 2 \, y - 17\]

Answer.

The isoclines cross at \(( -5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 63

S3: Phase planes (ver. 63)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + y + 12\]

\[y'= -3 \, x - 2 \, y + 4\]

Answer.

The isoclines cross at \(( 4 , -4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 64

S3: Phase planes (ver. 64)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x + y + 18\]

\[y'= -4 \, x - 3 \, y + 26\]

Answer.

The isoclines cross at \(( 5 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 65

S3: Phase planes (ver. 65)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x + 4 \, y - 7\]

\[y'= 2 \, x - 5 \, y - 16\]

Answer.

The isoclines cross at \(( 3 , -2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 66

S3: Phase planes (ver. 66)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - y - 3\]

\[y'= -3 \, x - 3 \, y - 15\]

Answer.

The isoclines cross at \(( -1 , -4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 67

S3: Phase planes (ver. 67)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 4 \, y + 25\]

\[y'= x - 5 \, y + 20\]

Answer.

The isoclines cross at \(( 5 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 68

S3: Phase planes (ver. 68)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 2 \, y + 20\]

\[y'= -4 \, x - 4 \, y\]

Answer.

The isoclines cross at \(( -4 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 69

S3: Phase planes (ver. 69)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 3 \, y + 4\]

\[y'= -x - 3 \, y + 2\]

Answer.

The isoclines cross at \(( -1 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 70

S3: Phase planes (ver. 70)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - y - 18\]

\[y'= -5 \, x - y + 18\]

Answer.

The isoclines cross at \(( 4 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 71

S3: Phase planes (ver. 71)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 2 \, y - 5\]

\[y'= -x - 4 \, y - 1\]

Answer.

The isoclines cross at \(( 3 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 72

S3: Phase planes (ver. 72)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 2 \, y - 17\]

\[y'= 2 \, x - 3 \, y + 6\]

Answer.

The isoclines cross at \(( 3 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 73

S3: Phase planes (ver. 73)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 3 \, y - 8\]

\[y'= -3 \, x - 2 \, y + 6\]

Answer.

The isoclines cross at \(( 2 , 0 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 74

S3: Phase planes (ver. 74)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 5 \, y + 13\]

\[y'= 3 \, x + 5 \, y - 1\]

Answer.

The isoclines cross at \(( -3 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 75

S3: Phase planes (ver. 75)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -3 \, x - 3 \, y - 9\]

\[y'= 2 \, x - 5 \, y - 1\]

Answer.

The isoclines cross at \(( -2 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 76

S3: Phase planes (ver. 76)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 5 \, y + 30\]

\[y'= 3 \, x - 3 \, y - 12\]

Answer.

The isoclines cross at \(( 5 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 77

S3: Phase planes (ver. 77)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 4 \, y - 4\]

\[y'= x + 2 \, y - 13\]

Answer.

The isoclines cross at \(( 5 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 78

S3: Phase planes (ver. 78)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 2 \, y + 13\]

\[y'= -2 \, x - 5 \, y + 19\]

Answer.

The isoclines cross at \(( -3 , 5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 79

S3: Phase planes (ver. 79)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x - 4 \, y - 12\]

\[y'= -x + 2 \, y + 2\]

Answer.

The isoclines cross at \(( -2 , -2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 80

S3: Phase planes (ver. 80)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - 5 \, y + 5\]

\[y'= -3 \, x - 3 \, y + 21\]

Answer.

The isoclines cross at \(( 3 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 81

S3: Phase planes (ver. 81)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - y - 12\]

\[y'= x + 2 \, y + 4\]

Answer.

The isoclines cross at \(( 4 , -4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 82

S3: Phase planes (ver. 82)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x - y - 18\]

\[y'= -2 \, x - 4 \, y - 6\]

Answer.

The isoclines cross at \(( 3 , -3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 83

S3: Phase planes (ver. 83)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -3 \, x - 2 \, y - 22\]

\[y'= -x + y + 1\]

Answer.

The isoclines cross at \(( -4 , -5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 84

S3: Phase planes (ver. 84)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 2 \, y - 6\]

\[y'= x - y - 2\]

Answer.

The isoclines cross at \(( 2 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 85

S3: Phase planes (ver. 85)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 5 \, x + 4 \, y + 5\]

\[y'= x - 3 \, y + 1\]

Answer.

The isoclines cross at \(( -1 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 86

S3: Phase planes (ver. 86)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 4 \, y - 8\]

\[y'= -4 \, x - 4 \, y - 32\]

Answer.

The isoclines cross at \(( -3 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 87

S3: Phase planes (ver. 87)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -x - 3 \, y + 14\]

\[y'= 4 \, x - 5 \, y + 29\]

Answer.

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 88

S3: Phase planes (ver. 88)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - 4 \, y + 3\]

\[y'= 5 \, x - 4 \, y - 27\]

Answer.

The isoclines cross at \(( 3 , -3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 89

S3: Phase planes (ver. 89)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - y - 15\]

\[y'= -4 \, x - 3 \, y + 7\]

Answer.

The isoclines cross at \(( 4 , -3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 90

S3: Phase planes (ver. 90)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -2 \, x + 3 \, y - 17\]

\[y'= -5 \, x - 2 \, y + 5\]

Answer.

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 91

S3: Phase planes (ver. 91)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 5 \, y - 31\]

\[y'= -4 \, x - 5 \, y - 17\]

Answer.

The isoclines cross at \(( 2 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 92

S3: Phase planes (ver. 92)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - y + 14\]

\[y'= -x - 5 \, y + 7\]

Answer.

The isoclines cross at \(( -3 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 93

S3: Phase planes (ver. 93)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -5 \, x - y - 20\]

\[y'= 2 \, x - y + 8\]

Answer.

The isoclines cross at \(( -4 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 94

S3: Phase planes (ver. 94)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x + 4 \, y + 3\]

\[y'= 5 \, x - 2 \, y - 21\]

Answer.

The isoclines cross at \(( 3 , -3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system.

Example 95

S3: Phase planes (ver. 95)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 3 \, y + 15\]

\[y'= 3 \, x + 5 \, y - 11\]

Answer.

The isoclines cross at \(( -3 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.

Example 96

S3: Phase planes (ver. 96)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - y + 10\]

\[y'= -4 \, x + 4 \, y\]

Answer.

The isoclines cross at \(( 2 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system.

Example 97

S3: Phase planes (ver. 97)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x - y - 16\]

\[y'= 4 \, x - 3 \, y + 16\]

Answer.

The isoclines cross at \(( -4 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.

Example 98

S3: Phase planes (ver. 98)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= x - 3 \, y - 12\]

\[y'= -3 \, x - 4 \, y - 29\]

Answer.

The isoclines cross at \(( -3 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.

Example 99

S3: Phase planes (ver. 99)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -3 \, x + 5 \, y - 10\]

\[y'= -5 \, x - 5 \, y - 30\]

Answer.

The isoclines cross at \(( -5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system.

Example 100

S3: Phase planes (ver. 100)

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 2 \, x - 2 \, y + 6\]

\[y'= -x - y - 1\]

Answer.

The isoclines cross at \(( -2 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system.