Question

Solve the differential equation ${ye}^{\frac{x}{y}}dx=\left({xe}^{\frac{x}{y}}+y^2\right)dy$.

Answer 1

Given: $y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy$

$\Rightarrow \frac{dx}{dy}=\frac{x{e}^{\frac{x}{y}}+y^2}{y{e}^{\frac{x}{y}}}$

$=\frac{x}{y}+\frac{y}{e^{\frac{x}{y}}}$

Let $x=vy$

$\therefore \frac{dx}{dy}=v+y\frac{dv}{dy}$

$\Rightarrow v+y\frac{dv}{dy}=v+\frac{y}{e^v}$

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

$\Rightarrow \int e^{v}dv=\int \frac{y}{y}dy$

$\Rightarrow e^v=y+C$

$\Rightarrow e^{\frac{x}{y}}=y+C$