Solve an Initial Value Problem Using a Green's Function

Solve an initial value problem for an inhomogeneous differential equation using GreenFunction.

First, compute the Green's function.

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gf[s_, t_] = GreenFunction[{-u''[t] + u'[t] - 37/4 u[t], u[0] == 0, u'[0] == 0}, u[t], {t, 0, \[Infinity]}, s]

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Define a forcing function.

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f[t_] := Cos[a t]

Convolving the Green's function with the forcing function yields the solution.

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sol = Integrate[gf[s, t] f[s], {s, 0, \[Infinity]}, Assumptions -> t > 0]

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Compare with the result given by DSolveValue.

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DSolveValue[{-u''[t] + u'[t] - 37/4 u[t] == f[t], u[0] == 0, u'[0] == 0}, u[t], t] // FullSimplify

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Plot the solution for different values of the parameter .

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Plot[Table[sol, {a, 1, 4, 0.8}] // Evaluate, {t, 3, 6}]

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