(1+ x ^2 #160; )dy = xydx ⇒ #160; dy y = xdx 1+x ^2 #160; #160; #160;Let 1+x ^2 #160; =t #160; #160; ⇒ 2xdx =dt ...

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Variable Separable Method

Solve - 1+x...

Question

Solve - $(1 + x^{2}) d y = x y d x$

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Solution

$(1 + x^{2}) d y = x y d x$

$\Rightarrow \frac{d y}{y} = \frac{x d x}{{1 + x}^{2}}$

Let ${1 + x}^{2} = t$
$\Rightarrow 2 x d x = d t$

$\Rightarrow \int \frac{d y}{y} = \frac{1}{2} \int \frac{d t}{t}$

$\Rightarrow ln | y | = \frac{1}{2} | t | + C$

$\Rightarrow y = e^{c} \sqrt{t}$

$\Rightarrow y = k \sqrt{{1 + x}^{2}}$

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S o l u t i o n o f t h e e q u a t i o n : (1 - x^{2}) d y + x y d x = x y^{2} d x

(1 - x^{2}) \frac{d y}{d x} + 2 x y = x {(1 - x^{2})}^{1 / 2}

x^{3} d y + x y d x = x^{2} d y + 2 y d x; y (2) = e

x > 1

y (4)

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