Question

True or false.

If $\frac{dy}{dx}+\frac{y^2}{x}=y$, then $kx=e^{1/v}=e^{x/y}$ (Enter 1 if true or 0 otherwise)

Answer 1

Given, $x\frac{dy}{dx}+\frac{y^2}{x}=y.$
$\frac{dy}{dx}+{\left(\frac{y}{x}\right)}^2=\frac{y}{x}$
Let $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
$\therefore v+x\frac{dv}{dx}+v^2=v\Rightarrow -\frac{dv}{v^2}=\frac{dx}{x}$
Integrating we get $\frac{1}{v}=\log x+\log k=\log \left(kx\right)$, where $k$ is integration constant.
$\Rightarrow kx=e^{1/v}=e^{x/y}$