Question

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

• A. $e^y=2e^x+\frac{x^3}{3}+C$
• B. $e^{-y}=2e^x+\frac{x^{-3}}{3}+C$
• C. $e^{-y}=2e^x+\frac{x^3}{3}+C$
• D. $e^y=2e^{-x}+\frac{x^3}{3}+C$

Answer 1

The correct option is A $e^y=2e^x+\frac{x^3}{3}+C$

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

$\Rightarrow \frac{dy}{dx}=2e^x\cdot \frac{1}{e^y}+x^2\cdot \frac{1}{e^y}$

$\Rightarrow \frac{dy}{dx}=\frac{1}{e^y}(2e^x+x^2)$

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

On integrating both sides, we get

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

$\Rightarrow e^y=2e^x+\frac{x^3}{3}+C$
which is the required solution.