Solve the following differential equation: y(1 - x^2) dy dx = x(1… - Vidyadip

Question

Solve the following differential equation: $y(1 - x^2) \frac{\text{dy}}{\text{dx}} = x(1 + y^2).$

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Answer

Writing $y(1 - x^2)$
$ \frac{\text{dy}}{\text{dx}} = x(1 + y^2)$ as
$\int\frac{\text{ydy}}{\text{1 + y}^{2}}=\int\frac{\text{x}}{\text{1 - x}^{2}}\text{dx}$
$\Rightarrow \log|1 + y^2| = - \log|1 - x^2| + \log C_1$
$\Rightarrow (1 + y^2)^{_{. }}(1 - x^2) = C.$

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