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

x e^{x} + x^{2} = C

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x e^{x} - y^{2} = C

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y e^{x} + x^{2} = C

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y e^{x} - x^{2} = C

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Solution

The correct option is C $y e^{x} + x^{2} = C$
$e^{x} d y + (y e^{x} + 2 x) d x = 0$
$e^{x} d y = - (y e^{x} + 2 x) d x$
$\frac{d y}{d x} = \frac{- (y e^{x} + 2 x)}{e^{x}}$

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

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

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

This is of the form $\frac{d y}{d x} + P y = Q$

where $P = 1$ and $Q = - \frac{2 x}{e^{x}}$

$I . F = e^{\int P d x}$
$= e^{\int 1 d x} = e^{x}$

Solution is :
$y e^{x} = \int \frac{- 2 x}{e^{x}} e^{x} d x + C$

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

Suggest Corrections

Similar questions

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

The general solution of the differential equation is

A. xe^y + x² = C

B. xe^y + y² = C

C. ye^x + x² = C

D. ye^y+ x² = C

e^{x} d y + (y e^{x} + 2 x) d x = 0

(1 + e^{\frac{x}{y}}) d x + (1 - \frac{x}{y}) e^{\frac{x}{y}} d y = 0

c

y e^{x / y} d x = (x e^{x / y} + y^{2} sin y) d y

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