...ordinary differential equation... y′′ −xy′+y=0 general solution ::=?? .m.n.1970.


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Question Number 118364 by mnjuly1970 last updated on 17/Oct/20

$. . . o r d i n a r y d i f f e r e n t i a l e q u a t i o n . . .$ $y^{″} - {x y}^{'} + y = 0$ $g e n e r a l s o l u t i o n ::= ? ?$ $. m . n . 1970 .$
	Answered by mindispower last updated on 17/Oct/20

	$w e c a n s e e e a s l y t h a t y = x i s s o l u t i o n$ $l e t y = x z$ $\Rightarrow y^{'} = {x z}^{'} + z$ $y^{″} = {x z}^{″} + 2 z^{'}$ ${x z}^{″} + 2 z^{'} - x ({x z}^{'} + z) + x z = 0$ $\Rightarrow {x z}^{″} + (2 - x^{2}) z^{'} = 0$ $\Rightarrow \int \frac{d (z^{'})}{z^{'}} = \int \frac{x^{2} - 2}{x} d x = (\frac{x^{2}}{2} - 2 l n (x))$ $\Rightarrow z = \frac{k}{x^{2}} e^{\frac{x^{2}}{2}}$ $b y p a r t z = (- \frac{k}{x} e^{\frac{x^{2}}{2}}) + \int {k e}^{\frac{x^{2}}{2}} d x$ $x = i y \sqrt{2} i n 2 n d$ $= z (x) = - \frac{k}{2} e^{\frac{x^{2}}{2}} + k \int i \sqrt{2} e^{- y^{2}} d y$ $= - \frac{k}{2} e^{\frac{x^{2}}{2}} + i k \frac{\sqrt{π}}{2} e r f (\frac{x}{i \sqrt{2}})$ $y (x) = x (- \frac{k}{2} e^{\frac{x^{2}}{2}} + i k \frac{\sqrt{π}}{2} e r f (\frac{x}{i \sqrt{2}}) + c)$
	Commented by mnjuly1970 last updated on 17/Oct/20

	$t h a n k y o u s o m u c h m r . . .$
	Commented by mindispower last updated on 19/Oct/20

	$w i t h e p l e a s u r s i r$
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