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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\frac{\text{y}}{\text{x}}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\text{dx}-\Big\{\frac{\text{x}}{\text{y}}\sin\Big(\frac{\text{y}}{\text{x}}\Big)+\cos\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}=0$
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Answer

Here, $\frac{\text{y}}{\text{x}}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\text{dx}-\Big\{\frac{\text{x}}{\text{y}}\sin\Big(\frac{\text{y}}{\text{x}}\Big)+\cos\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}=0$
$\frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{y}}{\text{x}}\cos\big(\frac{\text{y}}{\text{x}}\big)}{\frac{\text{x}}{\text{y}}\sin\big(\frac{\text{y}}{\text{x}}\big)+\cos\big(\frac{\text{y}}{\text{x}}\big)}$
It is a homogeneous equation
Put y = vx
and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\frac{\text{vx}}{\text{x}}\cos\big(\frac{\text{vx}}{\text{x}}\big)}{\frac{\text{x}}{\text{vx}}\sin\big(\frac{\text{vx}}{\text{x}}\big)+\cos\big(\frac{\text{vx}}{\text{x}}\big)}$
$=\frac{\text{v}\cos\text{v}}{\frac{1}{\text{v}}\sin\text{v}+\cos\text{v}}$
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2\cos\text{v}}{\sin\text{v}+\text{v}\cos\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2\cos\text{v}}{\sin\text{v}+\text{v}\cos\text{v}}-\text{v}$
$$$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2\cos\text{v}-\text{v}\sin\text{v}-\text{v}^2\cos\text{v}}{\sin\text{v}+\text{v}\cos\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{-\text{v}\sin\text{v}}{\sin\text{v}+\text{v}\cos\text{v}}$
$\frac{\sin\text{v}+\text{v}\cos\text{v}}{\text{v}\sin\text{v}}\text{dv}=-\frac{\text{dx}}{\text{x}}$
$\int\Big(\frac{1}{\text{v}}+\cot\text{v}\Big)\text{dv}=-\log|\text{x}|+\log|\text{C}|$
$\log|\text{v}|+\log|\sin\text{v}|=\log\Big|\frac{\text{C}}{\text{x}}\Big|$
$\log|\text{v}\sin\text{v}|=\log\Big|\frac{\text{C}}{\text{x}}\Big|$
$|\text{v}\sin\text{v}|=\Big|\frac{\text{C}}{\text{x}}\Big|$
$\Big|\text{x}\Big(\frac{\text{y}}{\text{x}}\Big)\sin\Big(\frac{\text{y}}{\text{x}}\Big)\Big|=|\text{C}|$
$\Big|\text{y}\sin\frac{\text{y}}{\text{x}}\Big|=\text{C}$

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