Solve the following differential equation: 2x^ 2 dy dx -2 xy + y^… - Vidyadip

Question

Solve the following differential equation:
$\text{2x}^{2}\frac{\text{dy}}{\text{dx}}-\text{2 xy + y}^{2}=0$

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Answer

$\text{2x}^{2}\frac{\text{dy}}{\text{dx}}-\text{2 xy + y}^{2}=0\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{2 xy - y}^{2}}{\text{2x}^{2}}=\frac{\text{2}\frac{\text{y}}{\text{x}}-\frac{\text{y}^{2}}{\text{x}^{2}}}{\text{2}}$Putting $\frac{\text{y}}{\text{x}}$ v so that y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v} + \text{x}\ \frac{\text{dv}}{\text{dx}}$
$\therefore\text{v + x }\frac{\text{dv}}{\text{dx}}=\text{v}-\frac{1}{2}\text{v}^{2}\therefore \text{x}\frac{\text{dv}}{\text{dx}}=-\frac{1}{2}\text{v}^{2}$
$\Rightarrow 2\int\frac{\text{dv}}{\text{v}^{2}} = -\int\frac{\text{dx}}{\text{x}}\Rightarrow\frac{2}{\text{v}}=\log\text{ x + c}$
$\therefore 2 \frac{\text{x}}{\text{y}}=\log\text{ x + c or y}=\frac{\text{2x}}{\text{log x + c}}.$

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