Chapter 10 - The Laplace Transform and Some Elementary Applications - 10.9 The Convolution Integral - Problems - Page 716: 29

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Given: $y''-ay=f(t)$ Since $y(0)=\alpha$ obtain: $L[y]=\frac{F(s)}{s^2-ay}+\frac{\alpha}{s^2-a}$ Using the Convolution Theorem $y(t)=L^{-1}[\frac{F(s)}{s^2-ay}]+L^{-1}[\frac{\alpha}{s^2-a}]\\ =\frac{1}{\sqrt a}\alpha\sin(at)+\int^t_0 \sin(\sqrt at)f(t-\tau)d \tau$