## 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 $$x( 0.80 )= -1.2$$.

$$-4$$ is a sink/stable. $$0$$ is a neither/unstable. $$2$$ is a source/unstable.

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

## 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 $$x( -1.8 )= 1.9$$.

$$-6$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 1.2 )= -0.90$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$4$$ is a sink/stable.

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

## 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 $$x( 0.80 )= -4.9$$.

$$-6$$ is a source/unstable. $$0$$ is a sink/stable. $$4$$ is a source/unstable.

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

## 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 $$x( 0.90 )= -3.1$$.

$$-5$$ is a source/unstable. $$0$$ is a neither/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( 1.8 )= -2.9$$.

$$-5$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 3.2 )= -1.8$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$6$$ is a source/unstable.

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

## 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 $$x( 0.80 )= -0.90$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$2$$ is a source/unstable.

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

## 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 $$x( -1.9 )= 2.2$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 1.1 )= -3.2$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$2$$ is a source/unstable.

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

## 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 $$x( -5.1 )= 0.90$$.

$$-6$$ is a sink/stable. $$0$$ is a source/unstable. $$2$$ is a sink/stable.

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

## 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 $$x( -1.1 )= 5.1$$.

$$-4$$ is a source/unstable. $$0$$ is a neither/unstable. $$6$$ is a sink/stable.

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

## 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 $$x( -4.2 )= 2.1$$.

$$-6$$ is a sink/stable. $$0$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( -1.1 )= 2.1$$.

$$-2$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( 1.1 )= -4.9$$.

$$-6$$ is a sink/stable. $$0$$ is a neither/unstable. $$5$$ is a source/unstable.

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

## 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 $$x( -1.2 )= 1.1$$.

$$-6$$ is a source/unstable. $$0$$ is a neither/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( -1.2 )= 4.8$$.

$$-4$$ is a sink/stable. $$0$$ is a source/unstable. $$6$$ is a sink/stable.

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

## 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 $$x( 0.80 )= -1.9$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( -0.90 )= 1.9$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$4$$ is a sink/stable.

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

## 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 $$x( 4.1 )= -3.2$$.

$$-4$$ is a source/unstable. $$0$$ is a neither/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 1.8 )= -2.9$$.

$$-5$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 1.1 )= -0.90$$.

$$-6$$ is a sink/stable. $$0$$ is a source/unstable. $$2$$ is a sink/stable.

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

## 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 $$x( 1.1 )= -4.9$$.

$$-6$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( 1.8 )= -3.1$$.

$$-5$$ is a sink/stable. $$0$$ is a source/unstable. $$4$$ is a sink/stable.

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

## 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 $$x( -1.1 )= 0.80$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 3.9 )= -0.80$$.

$$-6$$ is a sink/stable. $$0$$ is a neither/unstable. $$5$$ is a source/unstable.

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

## 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 $$x( -1.2 )= 1.1$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( -0.80 )= 1.9$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( 0.80 )= -2.2$$.

$$-5$$ is a sink/stable. $$0$$ is a neither/unstable. $$2$$ is a source/unstable.

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

## 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 $$x( 1.1 )= -4.8$$.

$$-6$$ is a sink/stable. $$0$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 0.90 )= -2.1$$.

$$-3$$ is a source/unstable. $$0$$ is a neither/unstable. $$2$$ is a sink/stable.

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

## 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 $$x( -0.90 )= 3.8$$.

$$-3$$ is a sink/stable. $$0$$ is a neither/unstable. $$6$$ is a source/unstable.

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

## 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 $$x( -0.90 )= 4.2$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 2.1 )= -1.1$$.

$$-5$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 3.1 )= -0.80$$.

$$-4$$ is a sink/stable. $$0$$ is a source/unstable. $$6$$ is a sink/stable.

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

## 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 $$x( -3.1 )= 1.8$$.

$$-4$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( -1.2 )= 1.9$$.

$$-2$$ is a sink/stable. $$0$$ is a neither/unstable. $$3$$ is a source/unstable.

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

## 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 $$x( -1.2 )= 0.80$$.

$$-6$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( -1.8 )= 0.80$$.

$$-3$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 1.1 )= -2.2$$.

$$-6$$ is a sink/stable. $$0$$ is a source/unstable. $$2$$ is a sink/stable.

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

## 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 $$x( -0.90 )= 2.1$$.

$$-2$$ is a sink/stable. $$0$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( -2.2 )= 0.90$$.

$$-3$$ is a sink/stable. $$0$$ is a source/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 3.8 )= -0.80$$.

$$-4$$ is a source/unstable. $$0$$ is a neither/unstable. $$5$$ is a sink/stable.

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

## 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 $$x( 0.80 )= -3.8$$.

$$-6$$ is a source/unstable. $$2$$ is a sink/stable.

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

## 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 $$x( -4.2 )= 4.1$$.

$$-6$$ is a sink/stable. $$0$$ is a neither/unstable. $$5$$ is a source/unstable.

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

## 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 $$x( 1.9 )= -3.2$$.

$$-4$$ is a sink/stable. $$0$$ is a source/unstable. $$3$$ is a sink/stable.

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

## 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 $$x( -4.2 )= 1.9$$.

$$-6$$ is a source/unstable. $$0$$ is a sink/stable. $$3$$ is a source/unstable.

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

## 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 $$x( 1.8 )= -1.2$$.

$$-3$$ is a source/unstable. $$0$$ is a neither/unstable. $$6$$ is a sink/stable.

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

## 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 $$x( -0.90 )= 1.2$$.

$$-5$$ is a source/unstable. $$4$$ is a sink/stable.

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

## 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 $$x( -1.1 )= 3.2$$.

$$-2$$ is a sink/stable. $$5$$ is a source/unstable.
$$\lim_{t\to\infty}x(t)=-2$$.