Answer to Question #165461 in Differential Equations for Nnnb

Question

x dy/dx = y + x

Answer 1

$xy'=y+x$

$y'=\frac{y}{x}+1$

Let $y=zx$, then $\frac{y}{x}=z$ and $y'=z+z'x$. Then we have

$z+z'x=z+1$, so $z'=\frac{1}{x}$

$z=\ln|x|+C$, so we have $y=zx=x\ln|x|+Cx$

Answer: $y=x\ln|x|+Cx$