Question

Solve the differential equation:
$x\frac{dy}{dx}-y\log x$, given that $y\left(1\right)=0$

Answer 1

$x\frac{dy}{dx}=ylogx$

$\frac{1}{y}dy=\frac{logx}{x}dx$ (integrating both)

$\int \frac{1}{y}dy=\int \frac{logx}{x}dx$

$logy=\frac{{log}^2\left(x\right)}{2}+C$

$y\left(1\right)=0$

$0=\frac{log\left(1\right)}{2}+C$

$C=0$

Hence

$logy=\frac{{log}^2\left(x\right)}{2}$.