Solve the following differential equation: (y2 - x2) dy = 3xy dx. - Vidyadip

Question

Solve the following differential equation:

(y² - x²) dy = 3xy dx.

✓

Answer

Writing $\frac{\text{dy}}{\text{dx}}=\frac{\text{3xy}}{\text{y}^{2}-\text{x}^{2}}=\frac{\text{3y/x}}{\text{y}^{2}/\text{x}^{2}-1}$
Putting $\frac{\text{y}}{\text{x}}=\text{v }\Rightarrow\text{v + x }\frac{\text{dv}}{\text{dx}}=\frac{\text{3v}}{\text{v}^{2}-1}\Rightarrow\text{ x }\frac{\text{dv}}{\text{dx}}=-\frac{\text{v}^{3}-\text{4v}}{\text{v}^{2}-1}$
$\therefore \int\frac{\text{v}^{2}-1}{\text{v}^{3}-\text{4v}}\text{ dv}=-\int\frac{\text{dx}}{\text{x}}\Rightarrow\frac{1}{8}\int\Bigg(\frac{2}{\text{v}}+\frac{3}{\text{v - 2}}+\frac{3}{\text{v + 2}}\Bigg)\text{dv}=-\int\frac{\text{dx}}{\text{x}}$
$\therefore $ 2 log v + 3 log (v - 2) + 3 log (v + 2) + 8 log x = log c
$\Rightarrow$ v² (v² - 4)³ x⁸ = c $\Rightarrow$ y² (y² - 4x²)³ = c.

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