First slide

Methods of solving first order first degree differential equations

Question

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

Moderate

A

$e^{- x^{2}} + e^{y^{2}} = c$
B

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

$e^{x^{2}} + e^{y^{2}} = c$
D

$e^{- x^{2}} + e^{- y^{2}} = c$

Solution

Given equation can be written as $y e^{- y^{2}} d y = x e^{x^{2}} d x$
( variables separable method)
$\Rightarrow \int 2 x e^{x^{2}} \cdot d x = - \int (- 2 y) e^{- y^{2}} \cdot d y$
$\Rightarrow \int e^{t} d t = - \int e^{u} \cdot d u where t = x^{2} and u = - y^{2}$
$\Rightarrow e^{x^{2}} = - e^{- y^{2}} + c$
$\Rightarrow e^{x^{2}} + e^{- y^{2}} = c$

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