Question

Solve the differential equation
$y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy(y\ne 0)$

Answer 1

Given, differential equation is $y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy$
$\Rightarrow e^{\frac{x}{y}}\frac{dx}{dy}=\frac{x}{y}{e}^{\frac{x}{y}}+y\Rightarrow e^{\frac{x}{y}}\frac{dx}{dy}-\frac{x}{y}{e}^{\frac{x}{y}}=y\dots \dots \left(i\right)$
On putting $\frac{x}{y}=v\Rightarrow x=vy\Rightarrow \frac{dx}{dy}=v+y\frac{dv}{dx}$
The Eq. (i) becomes
$e^v(v+y\frac{dv}{dy})-v{e}^v=y\Rightarrow e^v\frac{dv}{dy}=1\Rightarrow e^{v}dv=dy$
On integrating both sides, we get
$\int e^{v}dv=\int 1dy=e^v=y+C\Rightarrow e^{\frac{x}{y}}=y+C$