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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Find the particular solution of the differential equation $\text{x}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\frac{\text{dy}}{\text{dx}}=\text{y}\cos\Big(\frac{\text{y}}{\text{x}}\Big)+\text{x},$ given that when $\text{x}=1,\text{y}=\frac{\pi}4$.
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Answer

$\text{x}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\frac{\text{dy}}{\text{dx}}=\text{y}\cos\Big(\frac{\text{y}}{\text{x}}\Big)+\text{x}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{y}\cos\Big(\frac{\text{y}}{\text{x}}\Big)+\text{x}}{\text{x}\cos\Big(\frac{\text{y}}{\text{x}}\Big)}$
This is a homogeneous differential equation.
puttuing y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}},$ we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{vx}+\cos\text{v + x}}{\text{x}\cos\text{v}}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}+\cos\text{v + 1}}{\cos\text{v}}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}\cos\text{v}+1-\text{v}\cos\text{v}}{\cos{\text{v}}}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{1}{\cos\text{v}}$
$\Rightarrow\ \cos\text{v dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\cos\text{v dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \sin\text{v}=\log|\text{x}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}},$ we get
$\sin\frac{\text{y}}{\text{x}}=\log|\text{x}|+\text{C}\ \dots(1)$
At $\text{x}=1,\text{y}=\frac{\pi}4$ (Given)
Putting $\text{x}=1$ and $\text{y}=\frac{\pi}4$ in (1), we get
$\text{C}=\frac{1}{\sqrt2}$
Putting $\text{C}=\frac{1}{\sqrt2}$ in (1), we get
$\sin\frac{\text{y}}{\text{x}}=\log|\text{x}|+\frac{1}{\sqrt2}$
Hence, $\sin\frac{\text{y}}{\text{x}}=\log|\text{x}|+\frac{1}{\sqrt2}$ is the required solution.

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