**Systems of Equations Resulting from a Higher Order Equation**

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**with(DEtools):**

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**with(plots):**

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

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

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

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

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**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);**

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**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)});**

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

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**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**

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

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**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**

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

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**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**

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**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);**

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**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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