Symbolic & Numeric Calculus

Solve an ODE with a Ramp Forcing Function

Compute a pure ramp response.

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sol = DSolveValue[{x''[t] + a^2 x[t] == Ramp[t - a], x[0] == 0, x'[0] == 0}, x[t], t, Assumptions -> a > 0] // FullSimplify
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Plot the solution for different values of the parameter .

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Plot[Table[sol, {a, 1, 2, 0.3}] // Evaluate, {t, -1, 15}, Filling -> Axis]
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Combine the ramp with an impulsive force.

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sol = DSolveValue[{x''[t] + a^2 x[t] == Ramp[t - a] + DiracDelta[t - 2 a], x[0] == 0, x'[0] == 0}, x[t], t, Assumptions -> a > 0] // FullSimplify
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In[4]:=
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Plot[Table[sol, {a, 1, 2, 0.3}] // Evaluate, {t, -1, 15}, Filling -> Axis]
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