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.

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