Question

Solution of the differential equation
$y{e}^{x/y}dx=(x{e}^{x/y}+y^2\sin y)dy$, is

Answer 1

Given,

$y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2\sin y)dy$

$e^{\frac{x}{y}}dx=(\frac{x}{y}{e}^{\frac{x}{y}}+y\sin y)dy$

$\frac{dx}{dy}=\frac{x}{y}+\frac{y\sin y}{e^{\frac{x}{y}}}$

substitute $v=\frac{x}{y}\to \frac{dx}{dy}=ydvdy+v$

$y\frac{dv}{dy}+v=v+\frac{y\sin y}{e^v}$

$y\frac{dv}{dy}=\frac{y\sin y}{e^v}$

$e^{v}dv=\sin ydy$

integrating on both sides, $e^v=-\cos y+c$

$\therefore e^{\frac{x}{y}}+\cos y=c$