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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Solve the following differential equation:
$\text{x}^{2}\frac{\text{dy}}{\text{dx}}=\text{y}^{2}+\text{2xy}$
Given that y = 1, when x = 1.
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Answer

$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^{2}+\text{2xy}}{\text{x}^{2}}$
Let y = vx $\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\text{v}^{2}+\text{2v}\Rightarrow\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}^{2}+\text{v}$
$\Rightarrow\int\frac{\text{dv}}{\text{v(v+1)}}=\int\frac{\text{dx}}{\text{x}}$
$\Rightarrow\int\Bigg(\frac{1}{\text{v}}-\frac{1}{\text{v+1}}\Bigg)\text{dv}=\int\frac{\text{dx}}{\text{x}}$
$\log\frac{\text{v}}{\text{v+1}}=\log\text{cx}$
$\text{cx}=\frac{\frac{\text{y}}{\text{x}}}{\frac{\text{y}}{\text{x}}+1}=\frac{\text{y}}{\text{x+y}}$
When x = 1, y = 1, c = $\frac{1}{2}$
$\Rightarrow\frac{\text{x}}{2}=\frac{\text{y}}{\text{x+y}}\Rightarrow\text{x}^{2}+\text{xy - 2y}=0.$

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