solve-xy-x-2-1-y-x-2-sin-2x-

Question Number 115367 by Bird last updated on 25/Sep/20

s o l v e {x y}^{^{″}} - (x^{2} + 1) y^{^{'}} = x^{2} s i n (2 x)

Answered by Olaf last updated on 26/Sep/20

{x y}^{″} - (x^{2} + 1) y^{'} = x^{2} \sin 2 x

Let u = \frac{y^{'}}{x}

y^{″} = {x u}^{'} + u

x ({x u}^{'} + u) - (x^{2} + 1) x u = x^{2} \sin 2 x

{x u}^{'} + u - (x^{2} + 1) u = x \sin 2 x

{x u}^{'} - x^{2} u = x \sin 2 x

u^{'} - x u = \sin 2 x

e^{- \frac{x^{2}}{2}} u^{'} - {x e}^{- \frac{x^{2}}{2}} u = e^{- \frac{x^{2}}{2}} \frac{e^{i x} - e^{- i x}}{2 i}

\Rightarrow Integration by parts .

The function u is a linear combination

of functions \erf .

{It}^{'} s not a simple expression!

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