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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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