## 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)

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

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

#### 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)

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

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

#### 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)

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

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

#### 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)

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

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

#### 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)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 10

## S3: Phase planes (ver. 10)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 12

## S3: Phase planes (ver. 12)

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

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

#### 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)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 15

## S3: Phase planes (ver. 15)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 17

## S3: Phase planes (ver. 17)

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

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

#### Answer.

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

#### Example 18

## S3: Phase planes (ver. 18)

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

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

#### 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)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 21

## S3: Phase planes (ver. 21)

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

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

#### Answer.

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

#### Example 22

## S3: Phase planes (ver. 22)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 24

## S3: Phase planes (ver. 24)

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

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

#### Answer.

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

#### Example 25

## S3: Phase planes (ver. 25)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 27

## S3: Phase planes (ver. 27)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 29

## S3: Phase planes (ver. 29)

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

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

#### Answer.

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

#### Example 30

## S3: Phase planes (ver. 30)

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

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

#### Answer.

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

#### Example 31

## S3: Phase planes (ver. 31)

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

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

#### Answer.

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

#### Example 32

## S3: Phase planes (ver. 32)

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

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

#### Answer.

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

#### Example 33

## S3: Phase planes (ver. 33)

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

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

#### Answer.

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

#### Example 34

## S3: Phase planes (ver. 34)

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

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

#### Answer.

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

#### Example 35

## S3: Phase planes (ver. 35)

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

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

#### Answer.

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

#### Example 36

## S3: Phase planes (ver. 36)

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

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

#### Answer.

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

#### Example 37

## S3: Phase planes (ver. 37)

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

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

#### Answer.

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

#### Example 38

## S3: Phase planes (ver. 38)

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

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

#### Answer.

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

#### Example 39

## S3: Phase planes (ver. 39)

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

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

#### Answer.

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

#### Example 40

## S3: Phase planes (ver. 40)

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

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

#### Answer.

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

#### Example 41

## S3: Phase planes (ver. 41)

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

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

#### Answer.

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

#### Example 42

## S3: Phase planes (ver. 42)

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

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

#### Answer.

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

#### Example 43

## S3: Phase planes (ver. 43)

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

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

#### Answer.

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

#### Example 44

## S3: Phase planes (ver. 44)

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

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

#### Answer.

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

#### Example 45

## S3: Phase planes (ver. 45)

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

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

#### Answer.

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

#### Example 46

## S3: Phase planes (ver. 46)

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

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

#### Answer.

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

#### Example 47

## S3: Phase planes (ver. 47)

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

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

#### Answer.

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

#### Example 48

## S3: Phase planes (ver. 48)

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

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

#### Answer.

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

#### Example 49

## S3: Phase planes (ver. 49)

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

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

#### Answer.

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

#### Example 50

## S3: Phase planes (ver. 50)

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

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

#### Answer.

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

#### Example 51

## S3: Phase planes (ver. 51)

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

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

#### 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)

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

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

#### Answer.

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

#### Example 53

## S3: Phase planes (ver. 53)

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

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

#### Answer.

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

#### Example 54

## S3: Phase planes (ver. 54)

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

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

#### Answer.

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

#### Example 55

## S3: Phase planes (ver. 55)

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

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

#### Answer.

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

#### Example 56

## S3: Phase planes (ver. 56)

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

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

#### Answer.

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

#### Example 57

## S3: Phase planes (ver. 57)

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

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

#### Answer.

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

#### Example 58

## S3: Phase planes (ver. 58)

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

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

#### Answer.

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

#### Example 59

## S3: Phase planes (ver. 59)

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

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

#### Answer.

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

#### Example 60

## S3: Phase planes (ver. 60)

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

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

#### Answer.

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

#### Example 61

## S3: Phase planes (ver. 61)

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

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

#### Answer.

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

#### Example 62

## S3: Phase planes (ver. 62)

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

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

#### Answer.

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

#### Example 63

## S3: Phase planes (ver. 63)

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

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

#### Answer.

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

#### Example 64

## S3: Phase planes (ver. 64)

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

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

#### Answer.

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

#### Example 65

## S3: Phase planes (ver. 65)

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

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

#### Answer.

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

#### Example 66

## S3: Phase planes (ver. 66)

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

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

#### Answer.

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

#### Example 67

## S3: Phase planes (ver. 67)

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

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

#### Answer.

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

#### Example 68

## S3: Phase planes (ver. 68)

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

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

#### Answer.

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

#### Example 69

## S3: Phase planes (ver. 69)

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

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

#### Answer.

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

#### Example 70

## S3: Phase planes (ver. 70)

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

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

#### Answer.

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

#### Example 71

## S3: Phase planes (ver. 71)

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

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

#### Answer.

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

#### Example 72

## S3: Phase planes (ver. 72)

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

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

#### Answer.

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

#### Example 73

## S3: Phase planes (ver. 73)

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

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

#### Answer.

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

#### Example 74

## S3: Phase planes (ver. 74)

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

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

#### Answer.

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

#### Example 75

## S3: Phase planes (ver. 75)

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

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

#### Answer.

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

#### Example 76

## S3: Phase planes (ver. 76)

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

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

#### Answer.

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

#### Example 77

## S3: Phase planes (ver. 77)

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

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

#### Answer.

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

#### Example 78

## S3: Phase planes (ver. 78)

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

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

#### Answer.

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

#### Example 79

## S3: Phase planes (ver. 79)

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

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

#### Answer.

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

#### Example 80

## S3: Phase planes (ver. 80)

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

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

#### Answer.

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

#### Example 81

## S3: Phase planes (ver. 81)

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

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

#### Answer.

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

#### Example 82

## S3: Phase planes (ver. 82)

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

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

#### Answer.

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

#### Example 83

## S3: Phase planes (ver. 83)

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

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

#### Answer.

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

#### Example 84

## S3: Phase planes (ver. 84)

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

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

#### Answer.

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

#### Example 85

## S3: Phase planes (ver. 85)

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

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

#### Answer.

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

#### Example 86

## S3: Phase planes (ver. 86)

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

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

#### Answer.

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

#### Example 87

## S3: Phase planes (ver. 87)

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

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

#### Answer.

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

#### Example 88

## S3: Phase planes (ver. 88)

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

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

#### Answer.

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

#### Example 89

## S3: Phase planes (ver. 89)

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

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

#### Answer.

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

#### Example 90

## S3: Phase planes (ver. 90)

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

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

#### Answer.

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

#### Example 91

## S3: Phase planes (ver. 91)

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

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

#### Answer.

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

#### Example 92

## S3: Phase planes (ver. 92)

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

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

#### Answer.

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

#### Example 93

## S3: Phase planes (ver. 93)

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

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

#### Answer.

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

#### Example 94

## S3: Phase planes (ver. 94)

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

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

#### Answer.

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

#### Example 95

## S3: Phase planes (ver. 95)

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

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

#### Answer.

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

#### Example 96

## S3: Phase planes (ver. 96)

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

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

#### Answer.

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

#### Example 97

## S3: Phase planes (ver. 97)

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

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

#### Answer.

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

#### Example 98

## S3: Phase planes (ver. 98)

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

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

#### Answer.

#### Example 99

## S3: Phase planes (ver. 99)

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

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

#### Answer.

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

#### Example 100

## S3: Phase planes (ver. 100)

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

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

#### Answer.

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