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Variable Separable Method

Solve the dif...

Question

Solve the differential equation
$x y \frac{d y}{d x} = \frac{1 + y^{2}}{1 + x^{2}} (1 + x + x^{2})$

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Solution

Given,

$x y \frac{d y}{d x} = \frac{1 + y^{2}}{1 + x^{2}} (1 + x + x^{2})$
$\frac{y}{1 + y^{2}} d y = \frac{1 + x + x^{2}}{x (1 + x^{2})} d x$
$\frac{1}{2} \int \frac{2 y}{1 + y^{2}} d y = \int \frac{1}{x} d x + \int \frac{1}{1 + x^{2}} d x$
$\frac{1}{2} log | 1 + y^{2} | = log x + {tan}^{- 1} x + c$

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