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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equations:
$2\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2+\text{y}^2$
✓

Answer

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

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