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

Question

Solve the following differential equation:
$\big(\text{y}^2-2\text{xy}\big)\text{dx}=\big(\text{x}^2-2\text{xy}\big)\text{dy}$

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Answer

Here, $\big(\text{y}^2-2\text{xy}\big)\text{dx}=\big(\text{x}^2-2\text{xy}\big)\text{dy}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2-2\text{xy}}{\text{x}^2-2\text{xy}}$
It is a homogeneous equation.
Put y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2\text{x}^2-2\text{xvx}}{\text{x}^2-2\text{xvx}}$
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2-2\text{v}}{1-2\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2-2\text{v}}{1-2\text{v}}-\text{v}$
$=\frac{\text{v}^2-2\text{v}-\text{v}+2\text{v}^2}{1-2\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{3\text{v}^2-3\text{v}}{1-2\text{v}}$
$\frac{1-2\text{v}}{3(\text{v}^2-\text{v})}\text{dv}=\frac{\text{dx}}{\text{x}}$
$\frac{-(2\text{v}-1)}{3(\text{v}^2-\text{v})}\text{dv}=\frac{\text{dx}}{\text{x}}$
$\int\frac{2\text{v}-1}{\text{v}^2-\text{v}}\text{dv}=-3\int\frac{\text{dx}}{\text{x}}$
$\log\big|\text{v}^2-\text{v}\big|-3\log|\text{x}|+\log\text{C}$
$\text{v}^2-\text{v}=\frac{\text{C}}{\text{x}^3}$
$\frac{\text{y}^2}{\text{x}^2}-\frac{\text{y}}{\text{x}}=\frac{\text{C}}{\text{x}^3}$
$\text{y}^2-\text{xy}=\frac{\text{C}}{\text{x}}$
$\text{x}\big(\text{y}^2-\text{xy}\big)=\text{C}$

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