Question

Solve the differential equation $x+y\frac{dy}{dx}=2y$.

Answer 1

$x+y\frac{dy}{dx}=2y$.
$y\frac{dy}{dx}=2y-x$
$\frac{dy}{dx}=\frac{2y-x}{y}$
Let $y=vx$
$\frac{dy}{dx}=\frac{d}{dx}vx$
$=v\frac{d}{dx}x+x\frac{dv}{dx}$
$\frac{dy}{dx}=v+x\frac{dv}{dx}$
$v+x\frac{dv}{dx}=\frac{2vx-x}{vx}$
$x\frac{dv}{dx}=\frac{2vx-x}{vx}-v=-\frac{{(v-1)}^2}{v}$
$\int -\frac{{(v-1)}^2}{v}dv=\int \frac{1}{x}dx$
$\log x=-\int [\frac{1}{v-1}+\frac{1}{{(v-1)}^2}]dv=-[\log (v-1)-\frac{1}{v-1}]+c$
$\log x+\log (v-1)=\frac{1}{v-1}+c$
$\log x(v-1)=\frac{1}{v-1}+c$
$\log x(\frac{y}{x}-1)=\frac{1}{\frac{y}{x}-1}+c$
$\log (y-x)=\frac{x}{y-x}+c$