Find the particular solution of the differential equatione^ x 1 -… - Vidyadip

Question

Find the particular solution of the differential equation$\text{e}^{x}\sqrt{1 - \text{y}^{2}} \text{ dx} + \frac{\text{y}}{\text{x}}\text{dy } = 0 $ given that y= 1 when x=0.

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Answer

$\text{e}^{x}\sqrt{1 - \text{y}^{2}}\text{ dx} = \frac{-\text{y}}{\text{x}}\text{ dy } \Rightarrow\text{xe}^{x}\text{dx} = \frac{-\text{y}}{\sqrt{1 - \text{y}^{2}}}\text{ dy }$
Integrating both sides
$\int\text{xe}^{x}\text{ dx} = \frac{1}{2}\int-\frac{-2\text{y}}{\sqrt{1- \text{y}^{2}}}\text{ dy}$
$\Rightarrow\text{xe}^{x} - \text{e}^{x} =\sqrt{1 - \text{y}^{2}}+\text{c}$
For x = 0, y = 1, c = – 1 $\therefore\text{ solution is: } \text{ e}^{x} (\text{x} - 1 ) = \sqrt{1 - \text{y}^{2}} - 1 .$

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