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(D2-2d+1)y=exx2 |
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Answer» (D2-2D+1)y = exx2 It's auxiliary equation is m2-2m+1 = 0 ⇒ (m-1)2 = 0 ⇒ m = 1, 1 Therefore, C.F. = (C1+C2x)ex P.F. \(=\frac1{D^2-2D+1}\) exx2 \(=\frac1{(D-1)^2}\) exx2 = ex \(\frac1{(D+1-1)^2}x^2\) (∵ \(\frac1{f(D)}\) eaxV = eax \(\frac1{f(D+a)}V)\) = ex 1/D2 x2 = ex [∫(∫x2 dx)dx] (∵ 1/D V = ∫V dx) = ex(∫x3/3 dx) (∵ ∫xn dx \(=\frac{x^{n+1}}{n+1})\) \(=\frac{e^xx^4}{12}\) Hence, complete solution of given differential equation is y = C.F.+P.I. = (C1+C2x)ex \(+\frac{e^xx^4}{12}.\) |
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