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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
Solve the following differential equation:
$(y^2 - x^2) dy = 3xy\ dx$.
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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 $\frac{\text{y}}{\text{x}}=\text{v }\Rightarrow\text{v + x }\frac{\text{dv}}{\text{dx}}=\frac{\text{3v}}{\text{v}^{2}-1}\Rightarrow\text{ x }\frac{\text{dv}}{\text{dx}}=-\frac{\text{v}^{3}-\text{4v}}{\text{v}^{2}-1}$
$\therefore \int\frac{\text{v}^{2}-1}{\text{v}^{3}-\text{4v}}\text{ dv}=-\int\frac{\text{dx}}{\text{x}}\Rightarrow\frac{1}{8}\int\Bigg(\frac{2}{\text{v}}+\frac{3}{\text{v - 2}}+\frac{3}{\text{v + 2}}\Bigg)\text{dv}=-\int\frac{\text{dx}}{\text{x}}$
$\therefore 2 log\ v + 3 log\ (v - 2) + 3 log\ (v + 2) + 8 log\ x = log\ c$
$\Rightarrow v^2 (v^2 - 4)^3 x^8 = c$
$\Rightarrow y^2 (y^2 - 4x^2)^3 = c$.

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