Differential Equations Question 11
Question 11 - 2024 (29 Jan Shift 2)
If $\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$, then $\alpha^{2}$ is equal to
(1) 3
(2) 12
(3) 4
(4) 9
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Answer (1)
Solution
Differential equation :-
$x \cos \frac{y}{x} \frac{d y}{d x}=y \cos \frac{y}{x}+x$
$\cos \frac{y}{x}\left[x \frac{d y}{d x}-y\right]=x$
Divide both sides by $x^{2}$
$\cos \frac{y}{x}\left(\frac{x \frac{d y}{d x}-y}{x^{2}}\right)=\frac{1}{x}$
Let $\frac{y}{x}=t$
$\cos t\left(\frac{d t}{d x}\right)=\frac{1}{x}$
$\cos t d t=\frac{1}{x} d x$
Integrating both sides
$\sin t=\ln |x|+c$
$\sin \frac{y}{x}=\ln |x|+c$
Using $y(1)=\frac{\pi}{3}$, we get $c=\frac{\sqrt{3}}{2}$
So, $\alpha=\sqrt{3} \Rightarrow \alpha^{2}=3$
