Question

Solve $(1+e^{x/y})dx+e^{x/y}(1-\frac{x}{y})dy=0$ using $\frac{x}{y}=v$.

Answer 1

Given, $(1+e^{x/y})dx+e^{x/y}(1-\frac{x}{y})dy=0$
Use $\frac{x}{y}=v$
$x=v\cdot y$
$dx=vdy+dv\cdot y$
$(1+e^v)(vdy+dv\cdot y)+e^v(1-v)dy=0$
On simplification, we get
$dv\left(y\right)(1+e^v)+dy(v+e^v)=0$
$\Rightarrow \int dv\left(\frac{1+e^v}{v+e^v}\right)=-\int \frac{1}{y}dy$
$\Rightarrow \log |v+e^v|=-\log |y|+\log c$
$\Rightarrow \log |\frac{x}{y}+e^{x/y}||y|=\log c$
$\Rightarrow \left|(\frac{x}{y}+e^{x/y})\right(y\left)\right|=c$