The Solution of differential equation $\frac{{dy}}{{dx}} = {e^{x - y}} + {x^2}{e^{ - y}}$ is

Given that,

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

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

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

$\Rightarrow$ ${e^y}dy = \left( {{e^x} + {x^2}} \right)dx$

On integrating both sides,

we get
$\int {{e^y}} dy = \int {\left( {{e^x} + {x^2}} \right)} dx$

$\Rightarrow$ ${e^y} = {e^x} + \frac{{{x^3}}}{3} + C$

$\Rightarrow$ ${e^y} - {e^x} = \frac{{{x^3}}}{3} + C$