The general solution of the differential equation $e^{x} d y + (y e^{x} + 2 x) d x = 0$ is

a) $x e^{y} + x^{2} = C$

b) $x e^{y} + y^{2} = C$

c) $y e^{x} + x^{2} = C$

d) $y e^{y} + x^{2} = C$

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Solution

Given, differential equation is $e^{x} d y + (y e^{x} + 2 x) d x = 0$
$\Rightarrow e^{x} \frac{d y}{d x} + y e^{x} + 2 x = 0 \Rightarrow \frac{d y}{d x} + y = - 2 x e^{- x}$
This is a linear differential equation of the form
$\frac{d y}{d x} + P y = Q$
On comparing, $P = 1 a n d Q = - 2 x e^{- x} ∴ I F = e^{\int 1 d x} = e^{x}$
The generalsolution of the given differential equation is given by
$y . I F = \int (- 2 x e^{- x} \times e^{x}) d x + C \Rightarrow y e^{x} = - \int 2 x d x + C \Rightarrow y e^{x} = - x^{2} + C \Rightarrow y e^{x} + x^{2} + C,$ Hence, the correct option is (c).

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