Question

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

Answer 1

Given $(1+e^{x/y})dx+e^{x/y}(1-\frac{x}{y})dy=0$

$\Rightarrow (1+e^{x/y})\frac{dx}{dy}+e^{x/y}(1-\frac{x}{y})=0$

Put $x=vy\Rightarrow \frac{dx}{dy}=v+y\frac{dv}{dy}$

$(1+e^v)(v+y\frac{dv}{dy})+e^v(1-v)=0$

$\Rightarrow \frac{dy}{y}=\frac{-(1+e^v)}{(v+e^v)}dv$

Integrating, we get

ln $y=-ln(v+e^v)+lnc$

$\Rightarrow ln\left[y(\frac{x}{y}+e^{\frac{x}{y}})\right]=lnc$

$\Rightarrow x+y{e}^{x/y}=c$