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

Question

Solve the following differential equation:$\text{x }\frac{\text{dy}}{\text{dx}}=\text{y - x}\tan\Bigg(\frac{\text{y}}{\text{ax}}\Bigg).$

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Answer

$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\tan\Bigg(\frac{\text{y}}{\text{x}}\Bigg).........\text{(i)}$ Let y = vx $\Rightarrow$ $\frac{\text{dy}}{\text{dx}}=\text{v + x }\frac{\text{dv}}{\text{dx}}$ $\therefore\text{(i) becomes v + x }\frac{\text{dv}}{\text{dx}}=\text{v - tan v}$$\Rightarrow-\cot\text{v dv}=\frac{\text{dx}}{\text{x}}$
log | cosec v | = log | cx |
$\Rightarrow\text{ c x }=\text{cosec }\Bigg(\frac{\text{y}}{\text{x}}\Bigg)$
$\text{OR }\Bigg(\text{x sin}\Bigg(\frac{\text{y}}{\text{x}}\Bigg)=\text{c}\Bigg).$

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