solve-the-first-order-differential-equation-xdy-ydx-xy-1-2-dx-

Question Number 206449 by necx122 last updated on 15/Apr/24

s o l v e t h e f i r s t o r d e r d i f f e r e n t i a l

e q u a t i o n :

x d y - y d x = {(x y)}^{1 / 2} d x

Answered by Berbere last updated on 17/Apr/24

y^{'} x - y = \sqrt{x y}; y^{'} = \frac{d y}{d x}

\frac{y^{'} x - y}{x^{2}} = \sqrt{\frac{y}{x}} . \frac{1}{x}

d (\frac{y}{x}) = {(\frac{y}{x})}^{\frac{1}{2}} . \frac{1}{x} d x

\frac{d (\frac{y}{x})}{\sqrt{\frac{y}{x}}} = \frac{1}{x} \Rightarrow 2 \sqrt{\frac{y}{x}} = l n (x) + c \Rightarrow \frac{y}{x} = {(\frac{l n (x) + c}{2})}^{2}

y = x {(c^{'} + l n (\sqrt{x}))}^{2}

Commented by necx122 last updated on 17/Apr/24

p l e a s e, c s n y o u e x p l a i n t h e 3 r d l i n e ?

d i d y o u d o d y / d x o r d (y / x) ?

Commented by Berbere last updated on 17/Apr/24

\frac{y^{'} . x - 1 . y}{x^{2}} = {(\frac{y}{x})}^{'} = \frac{d}{d x} (\frac{y}{x})

Commented by necx122 last updated on 17/Apr/24

Thank you sir. It's clear now.

Commented by Berbere last updated on 18/Apr/24

w i t h e P l e a s u r

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