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Find:(D2 + 3D + 2)y = e2x-sin x |
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Answer» (D2 + 3D + 2)y = e2x-sin x m2 + 3m + 2 = 0 is its auxiliary equation ⇒ (m+ 1)(m + 2) = 0 ⇒ m = -1, -2 ∴ C.F. = C1e-x + C2e-2x P.I. = \(\frac1{D^2+3D+2}e^{2x}-\frac1{D^2+3D+2}sin x\) = \(\frac{e^{2x}}{12}-\frac{sin x}{-1+3D+2}\) = \(\frac{e^{2x}}{12}-\frac{sin x}{3D+1}\) = \(\frac{e^{2x}}{12}-\frac{(3D-1)}{9D^2-1}sin x\) = \(\frac{e^{2x}}{12}-\frac{(3D-1)sin x}{-9-1}\) = \(\frac{e^{2x}}{12}-\frac{3cos x-sin x}{-10}\) = \(\frac{e^{2x}}{12}+\frac3{10}cos x-\frac3{10} sin x\) |
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