The general solution of d y d x = 2 x e x 2 - y is:

Question

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

Smart Achievers · Accepted Answer

(3)

Ans. $e^{y} = e^{x^{2}} + C$

Explanation:

Given that,

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

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

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

On integrating both sides, we get

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

Put $x^{2} = t$ in RHS integral, we get

$2 x d x = d t$

$∴$ $\int e^{y} d y = \int e^{t} d t$

$\Rightarrow$ $e^{y} = e^{t} + C$

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

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