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X^2 d^2/dx^2+ x dy/dx - y = x^2 + 1 |
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Answer» Let x = ez \(\therefore x \frac{dy}{dx} = Dy\), where \(D = \frac d{dz}\) \(x^2\frac{d^2y}{dx^2} = D(D - 1)y\), where \(D = \frac d{dz}\) ∴ Given differential equation converts into D(D - 1)y + Dy - y = e2z + 1 It's auxiliary equation is m(m - 1) + m - 1 = 0 ⇒ m2 - 1 = 0 ⇒ m = \(\pm \)1 ∴ \(C.F. = C_1e^z + C_2e^{-z} = C_1x + \frac{C_2}x\) (∵ \(e^z = x\)) \(P.I. = \frac1{D^2 - 1} (e^{2z} + 1)\) \(= \frac1{D^2 - 1}e^{2z} + \frac1{D^2 - 1}e^{0z}\) \(= \frac{e^{2z}}{4 - 1} + \frac{e^{0z}}{0 - 1}\) \(= \frac{e^{2z}}{3} + \frac1{-1}\) \(= \frac{e^{2z}}{3} - 1\) \(= \frac{x^2}{3} - 1\) (∵ \(e^z = x\)) ∴ \(y = C.F. + P.I.\) \(= C_1x+ \frac{C_2}x + \frac{x^2}3 - 1\) which is complete solution of given differential equation. |
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