Systems of Equations Resulting from a Higher Order Equation

> with(DEtools):

> with(plots):

2nd Order Equation --> Systems of Two 1st Order Equations

> ode := diff(y(t), t$2) + .2*diff(y(t), t) + 1.01*y(t) = 0;

> odesys := diff(y(t), t) = v(t), diff(v(t), t) = -.2*v(t) - 1.01*y(t);

>

Plot of Solution (y(t)) for the 2nd Order Equation and Y-Component Curve for the System

> DEplot(ode, [y(t)], t = 0..30, [[y(0) = 0, D(y)(0) = -1], [y(0) = -1, D(y)(0) = 0]], stepsize = .1, linecolor = blue);

> display({DEplot(ode, [y(t)], t = 0..30, [[y(0) = 0, D(y)(0) = -1]], stepsize = .1, linecolor = blue), DEplot(ode, [y(t)], t = 0..30, [[y(0) = -1, D(y)(0) = 0]], stepsize = .1, linecolor = red)});

> DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1], [0, -1, 0]], stepsize = .1, linecolor = blue, scene = [t, y(t)]);

> display({DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1]], stepsize = .1, linecolor = blue, scene = [t, y(t)]), DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, -1, 0]], stepsize = .1, linecolor = red, scene = [t, y(t)])});

Advantages of System

V-Component Curve

> DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1], [0, -1, 0]], stepsize = .1, linecolor = blue, scene = [t, v(t)]);

> display({DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1]], stepsize = .1, linecolor = blue, scene = [t, v(t)]), DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, -1, 0]], stepsize = .1, linecolor = red, scene = [t, v(t)])});

Orbits - Trajectories - Phase Portrait

> DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1], [0, -1, 0]], stepsize = .1, linecolor = blue, scene = [y(t), v(t)]);

> display({DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1]], stepsize = .1, linecolor = blue, scene = [y(t), v(t)], color = wheat), DEplot({odesys}, [y(t), v(t)], t = 0..30, [[0, -1, 0]], stepsize = .1, linecolor = red, scene = [y(t), v(t)], color = gray)});

Time-State Graph

> DEplot3d({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1], [0, -1, 0]], stepsize = .1, linecolor = blue, scene = [y(t), v(t), t], axes = framed);

> display({DEplot3d({odesys}, [y(t), v(t)], t = 0..30, [[0, 0, -1]], stepsize = .1, linecolor = blue, scene = [y(t), v(t), t]), DEplot3d({odesys}, [y(t), v(t)], t = 0..30, [[0, -1, 0]], stepsize = .1, linecolor = red, scene = [y(t), v(t), t])}, axes = framed);

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