## F4: Autonomous ODEs

#### Example 1

## F4: Autonomous ODEs (ver. 1)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + 2 \, x^{3} - 8 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -1.2\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a neither/unstable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 2

## F4: Autonomous ODEs (ver. 2)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + 3 \, x^{2} - 18 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.8 )= 1.9\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 3

## F4: Autonomous ODEs (ver. 3)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 8 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.2 )= -0.90\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-2\).

#### Example 4

## F4: Autonomous ODEs (ver. 4)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + 2 \, x^{2} - 24 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -4.9\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 5

## F4: Autonomous ODEs (ver. 5)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - 2 \, x^{3} + 15 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -3.1\).

#### Answer.

\(-5\) is a source/unstable. \(0\) is a neither/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 6

## F4: Autonomous ODEs (ver. 6)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} + 2 \, x^{4} - 15 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -2.9\).

#### Answer.

\(-5\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 7

## F4: Autonomous ODEs (ver. 7)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} - 2 \, x^{4} - 24 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.2 )= -1.8\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(6\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 8

## F4: Autonomous ODEs (ver. 8)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} + 2 \, x^{4} - 8 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -0.90\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 9

## F4: Autonomous ODEs (ver. 9)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + x^{2} - 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.9 )= 2.2\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 10

## F4: Autonomous ODEs (ver. 10)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + 2 \, x^{2} - 8 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -3.2\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 11

## F4: Autonomous ODEs (ver. 11)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - 4 \, x^{2} + 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -5.1 )= 0.90\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=2\).

#### Example 12

## F4: Autonomous ODEs (ver. 12)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} + 2 \, x^{3} + 24 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 5.1\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a neither/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=6\).

#### Example 13

## F4: Autonomous ODEs (ver. 13)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 30 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -4.2 )= 2.1\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 14

## F4: Autonomous ODEs (ver. 14)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + x + 6\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 2.1\).

#### Answer.

\(-2\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 15

## F4: Autonomous ODEs (ver. 15)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + x^{3} - 30 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -4.9\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 16

## F4: Autonomous ODEs (ver. 16)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - x^{3} + 30 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 1.1\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a neither/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 17

## F4: Autonomous ODEs (ver. 17)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + 2 \, x^{2} + 24 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 4.8\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=6\).

#### Example 18

## F4: Autonomous ODEs (ver. 18)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + x^{2} - 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -1.9\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 19

## F4: Autonomous ODEs (ver. 19)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 8 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 1.9\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 20

## F4: Autonomous ODEs (ver. 20)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} + x^{3} + 20 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 4.1 )= -3.2\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a neither/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 21

## F4: Autonomous ODEs (ver. 21)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + 25\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -2.9\).

#### Answer.

\(-5\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 22

## F4: Autonomous ODEs (ver. 22)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - 4 \, x^{2} + 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -0.90\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 23

## F4: Autonomous ODEs (ver. 23)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} - 3 \, x + 18\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -4.9\).

#### Answer.

\(-6\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 24

## F4: Autonomous ODEs (ver. 24)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 20 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -3.1\).

#### Answer.

\(-5\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 25

## F4: Autonomous ODEs (ver. 25)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 3 \, x^{4} + 10 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 0.80\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 26

## F4: Autonomous ODEs (ver. 26)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + x^{3} - 30 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.9 )= -0.80\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 27

## F4: Autonomous ODEs (ver. 27)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + x^{4} + 6 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 1.1\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 28

## F4: Autonomous ODEs (ver. 28)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + x^{4} + 6 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.80 )= 1.9\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 29

## F4: Autonomous ODEs (ver. 29)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + 3 \, x^{3} - 10 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -2.2\).

#### Answer.

\(-5\) is a sink/stable. \(0\) is a neither/unstable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 30

## F4: Autonomous ODEs (ver. 30)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 30 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -4.8\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 31

## F4: Autonomous ODEs (ver. 31)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - x^{3} + 6 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -2.1\).

#### Answer.

\(-3\) is a source/unstable. \(0\) is a neither/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 32

## F4: Autonomous ODEs (ver. 32)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - 3 \, x^{3} - 18 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 3.8\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a neither/unstable. \(6\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 33

## F4: Autonomous ODEs (ver. 33)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + 3 \, x^{2} + 10 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 4.2\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 34

## F4: Autonomous ODEs (ver. 34)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} + 2 \, x^{4} - 15 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.1 )= -1.1\).

#### Answer.

\(-5\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 35

## F4: Autonomous ODEs (ver. 35)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 24 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.1 )= -0.80\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 36

## F4: Autonomous ODEs (ver. 36)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + x^{2} - 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -3.1 )= 1.8\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 37

## F4: Autonomous ODEs (ver. 37)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - x^{3} - 6 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 1.9\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a neither/unstable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 38

## F4: Autonomous ODEs (ver. 38)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} - x + 30\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 0.80\).

#### Answer.

\(-6\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 39

## F4: Autonomous ODEs (ver. 39)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - 9\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.8 )= 0.80\).

#### Answer.

\(-3\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 40

## F4: Autonomous ODEs (ver. 40)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - 4 \, x^{2} + 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -2.2\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 41

## F4: Autonomous ODEs (ver. 41)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + x^{4} + 6 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 2.1\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 42

## F4: Autonomous ODEs (ver. 42)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 15 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -2.2 )= 0.90\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 43

## F4: Autonomous ODEs (ver. 43)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} + x^{3} + 20 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.8 )= -0.80\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a neither/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 44

## F4: Autonomous ODEs (ver. 44)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} - 4 \, x + 12\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -3.8\).

#### Answer.

\(-6\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=2\).

#### Example 45

## F4: Autonomous ODEs (ver. 45)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + x^{3} - 30 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -4.2 )= 4.1\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 46

## F4: Autonomous ODEs (ver. 46)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 12 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.9 )= -3.2\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 47

## F4: Autonomous ODEs (ver. 47)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + 3 \, x^{2} - 18 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -4.2 )= 1.9\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 48

## F4: Autonomous ODEs (ver. 48)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} + 3 \, x^{3} + 18 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -1.2\).

#### Answer.

\(-3\) is a source/unstable. \(0\) is a neither/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 49

## F4: Autonomous ODEs (ver. 49)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} - x + 20\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 1.2\).

#### Answer.

\(-5\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 50

## F4: Autonomous ODEs (ver. 50)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - 3 \, x - 10\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 3.2\).

#### Answer.

\(-2\) is a sink/stable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-2\).

#### Example 51

## F4: Autonomous ODEs (ver. 51)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + x^{2} + 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.1 )= -1.2\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 52

## F4: Autonomous ODEs (ver. 52)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 24 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -1.9\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 53

## F4: Autonomous ODEs (ver. 53)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - 4 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -0.90\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a neither/unstable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-2\).

#### Example 54

## F4: Autonomous ODEs (ver. 54)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - x^{3} - 20 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.1 )= -2.9\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 55

## F4: Autonomous ODEs (ver. 55)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} + 3 \, x^{4} - 18 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.2 )= -2.8\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 56

## F4: Autonomous ODEs (ver. 56)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - x - 12\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 1.9\).

#### Answer.

\(-3\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 57

## F4: Autonomous ODEs (ver. 57)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - 2 \, x - 15\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 3.1 )= -0.90\).

#### Answer.

\(-3\) is a sink/stable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 58

## F4: Autonomous ODEs (ver. 58)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + 3 \, x^{2} - 10 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -4.2 )= 1.2\).

#### Answer.

\(-5\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 59

## F4: Autonomous ODEs (ver. 59)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + 9\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -2.2\).

#### Answer.

\(-3\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 60

## F4: Autonomous ODEs (ver. 60)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - 3 \, x^{2} + 18 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -0.90\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-6\).

#### Example 61

## F4: Autonomous ODEs (ver. 61)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + x^{4} + 30 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 5.1 )= -2.1\).

#### Answer.

\(-5\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 62

## F4: Autonomous ODEs (ver. 62)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - x^{2} + 20 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -3.2 )= 2.8\).

#### Answer.

\(-5\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 63

## F4: Autonomous ODEs (ver. 63)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} + x - 20\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -3.2 )= 1.9\).

#### Answer.

\(-5\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 64

## F4: Autonomous ODEs (ver. 64)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} + x^{3} - 30 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -2.2 )= 1.8\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 65

## F4: Autonomous ODEs (ver. 65)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + x^{2} - 12 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -0.90\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 66

## F4: Autonomous ODEs (ver. 66)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + 4\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 1.2\).

#### Answer.

\(-2\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=2\).

#### Example 67

## F4: Autonomous ODEs (ver. 67)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} - 4 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -0.90\).

#### Answer.

\(-2\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 68

## F4: Autonomous ODEs (ver. 68)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 8 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 3.1\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 69

## F4: Autonomous ODEs (ver. 69)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - 2 \, x^{3} + 8 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -3.1\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a neither/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 70

## F4: Autonomous ODEs (ver. 70)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} - 36 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -2.2 )= 3.9\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(6\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 71

## F4: Autonomous ODEs (ver. 71)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} - 4 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.90 )= 1.2\).

#### Answer.

\(-2\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 72

## F4: Autonomous ODEs (ver. 72)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} - 9 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -2.1 )= 0.90\).

#### Answer.

\(-3\) is a source/unstable. \(0\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 73

## F4: Autonomous ODEs (ver. 73)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - 16 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 2.8\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a neither/unstable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 74

## F4: Autonomous ODEs (ver. 74)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - 2 \, x^{3} + 8 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.2 )= -1.8\).

#### Answer.

\(-4\) is a source/unstable. \(0\) is a neither/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 75

## F4: Autonomous ODEs (ver. 75)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + x^{2} + 6 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -1.1\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-2\).

#### Example 76

## F4: Autonomous ODEs (ver. 76)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - x^{3} - 12 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.80 )= 2.9\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a neither/unstable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 77

## F4: Autonomous ODEs (ver. 77)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} - 3 \, x^{4} - 10 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.2 )= -1.1\).

#### Answer.

\(-2\) is a source/unstable. \(0\) is a sink/stable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 78

## F4: Autonomous ODEs (ver. 78)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 6 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.8 )= 1.1\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=2\).

#### Example 79

## F4: Autonomous ODEs (ver. 79)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - 2 \, x^{2} + 15 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -4.2\).

#### Answer.

\(-5\) is a sink/stable. \(0\) is a source/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 80

## F4: Autonomous ODEs (ver. 80)

Draw a phase line for the following autonomous ODE.

\[x'= x^{5} + 2 \, x^{4} - 24 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.80 )= 2.9\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 81

## F4: Autonomous ODEs (ver. 81)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - 9\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -2.2 )= 1.1\).

#### Answer.

\(-3\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 82

## F4: Autonomous ODEs (ver. 82)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} + 2 \, x - 15\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.9 )= -1.2\).

#### Answer.

\(-5\) is a sink/stable. \(3\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 83

## F4: Autonomous ODEs (ver. 83)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + 16\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -3.2 )= 2.2\).

#### Answer.

\(-4\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 84

## F4: Autonomous ODEs (ver. 84)

Draw a phase line for the following autonomous ODE.

\[x'= x^{3} + x^{2} - 6 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 0.80\).

#### Answer.

\(-3\) is a source/unstable. \(0\) is a sink/stable. \(2\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 85

## F4: Autonomous ODEs (ver. 85)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} - x^{2} + 6 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.80 )= -0.90\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 86

## F4: Autonomous ODEs (ver. 86)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} + 25\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.8 )= -0.80\).

#### Answer.

\(-5\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=5\).

#### Example 87

## F4: Autonomous ODEs (ver. 87)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 2 \, x^{4} + 15 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.9 )= -2.1\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 88

## F4: Autonomous ODEs (ver. 88)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - x - 30\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 4.2 )= -0.90\).

#### Answer.

\(-5\) is a sink/stable. \(6\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-5\).

#### Example 89

## F4: Autonomous ODEs (ver. 89)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - x - 12\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.1 )= 2.1\).

#### Answer.

\(-3\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 90

## F4: Autonomous ODEs (ver. 90)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} + 36 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -3.9 )= 2.8\).

#### Answer.

\(-6\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=6\).

#### Example 91

## F4: Autonomous ODEs (ver. 91)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - x - 12\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.2 )= -0.80\).

#### Answer.

\(-3\) is a sink/stable. \(4\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 92

## F4: Autonomous ODEs (ver. 92)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - 4 \, x^{3} + 12 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 0.90 )= -2.2\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a neither/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 93

## F4: Autonomous ODEs (ver. 93)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} + 9 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.2 )= 1.9\).

#### Answer.

\(-3\) is a source/unstable. \(0\) is a neither/unstable. \(3\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=3\).

#### Example 94

## F4: Autonomous ODEs (ver. 94)

Draw a phase line for the following autonomous ODE.

\[x'= x^{2} - 2 \, x - 24\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -1.8 )= 1.9\).

#### Answer.

\(-4\) is a sink/stable. \(6\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 95

## F4: Autonomous ODEs (ver. 95)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{5} - x^{4} + 6 \, x^{3}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.1 )= -1.8\).

#### Answer.

\(-3\) is a sink/stable. \(0\) is a source/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-3\).

#### Example 96

## F4: Autonomous ODEs (ver. 96)

Draw a phase line for the following autonomous ODE.

\[x'= x^{4} - x^{3} - 20 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -0.80 )= 1.1\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a neither/unstable. \(5\) is a source/unstable.

\(\lim_{t\to\infty}x(t)=0\).

#### Example 97

## F4: Autonomous ODEs (ver. 97)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + 3 \, x^{2} + 10 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.9 )= -1.2\).

#### Answer.

\(-2\) is a sink/stable. \(0\) is a source/unstable. \(5\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-2\).

#### Example 98

## F4: Autonomous ODEs (ver. 98)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{2} - x + 20\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 1.8 )= -3.9\).

#### Answer.

\(-5\) is a source/unstable. \(4\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=4\).

#### Example 99

## F4: Autonomous ODEs (ver. 99)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{3} + 2 \, x^{2} + 24 \, x\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( 2.9 )= -1.8\).

#### Answer.

\(-4\) is a sink/stable. \(0\) is a source/unstable. \(6\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=-4\).

#### Example 100

## F4: Autonomous ODEs (ver. 100)

Draw a phase line for the following autonomous ODE.

\[x'= -x^{4} - 4 \, x^{3} + 12 \, x^{2}\]

Label each equilibrium of the ODE as stable/unstable and sink/source/neither. Then compute \(\lim_{t\to\infty}x(t)\) assuming the initial condition \(x( -4.1 )= 0.80\).

#### Answer.

\(-6\) is a source/unstable. \(0\) is a neither/unstable. \(2\) is a sink/stable.

\(\lim_{t\to\infty}x(t)=2\).