The solution of differential equation d y d x = e x - y + x 2 e - y is:

Question

The solution of differential equation $\frac{d y}{d x} = e^{x - y} + x^{2} e^{- y}$ is:

Smart Achievers · Accepted Answer

(2)

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

Explanation:

Given that, $\frac{d y}{d x} = e^{x - y} + x^{2} e^{- y}$

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

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

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

On integrating both sides, we get

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

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

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

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