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

Question

Solve the following differential equation:

$(\text{x}^{2} - 1 ) \frac{\text{dy}}{\text{dx}} + 2 \text{xy} = \frac{2}{\text{x}^{2} - 1 }.$

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Answer

Given differential equation can be written as
$\frac{\text{dy}}{\text{dx}} + \frac{2\text{x}}{\text{x}^{2} - 1 }\text{y} = \frac{2}{(\text{x}^{2} - 1 )^{2}}$
Integrating factor = $\text{e}^{\int\frac{2\text{x}}{\text{x}^{2} - 1}\text{dx}} = \text{e}^{\log(\text{x}^{2} - 1 )} = \text{x}^{2} - 1 $
$\therefore\text{ Solution is }\text{y}.(\text{x}^{2} - 1 ) =\int\frac{2}{(\text{x}^{2} - 1 )^{2}}.(\text{x}^{2} - 1 )\text{dx} + \text{c}$
$\Rightarrow\text{y}(\text{x}^{2} - 1 ) = 2 \int\frac{1}{\text{x}^{2} - 1}\text{ dx} + \text{c} $
$\Rightarrow\text{y}(\text{x}^{2} - 1 ) = \log\bigg|\frac{\text{x} - 1}{\text{x} + 1 }\bigg| + \text{c}.$

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