If ((y)/(x))= _e|x|+( )/(2) is the solution of the differential equation x ((y)/(x))(dy)/(…

Question

If $\sin (\frac{y}{x})={\log }_{e}|x|+\frac{\alpha }{2}$ is the solution of the differential equation $x\cos (\frac{y}{x})\frac{dy}{dx}=y\cos (\frac{y}{x})+x$ and $y(1)=\frac{\pi }{3}$, then ${\alpha }^{2}$ is equal to

Accepted Answer

Differential equation :-

$\begin{gathered} x\cos \frac{y}{x}\frac{dy}{dx}=y\cos \frac{y}{x}+x \ \cos \frac{y}{x}[x\frac{dy}{dx}-y]=x \end{gathered}$

Divide both sides by ${x}^{2}$

$\cos \frac{y}{x}(\frac{x\frac{dy}{dx}-y}{{x}^{2}})=\frac{1}{x}$

Let $\frac{y}{x}=t$

$\begin{gathered} \cos t(\frac{dt}{dx})=\frac{1}{x} \ \cos tdt=\frac{1}{x}dx \end{gathered}$

Integrating both sides

$\begin{gathered} \sin t=\ln |x|+c \ \sin \frac{y}{x}=\ln |x|+c \end{gathered}$

Using $y(1)=\frac{\pi }{3}$, we get $c=\frac{\sqrt{3}}{2}$

So, $\alpha =\sqrt{3}\Rightarrow {\alpha }^{2}=3$

Solutions · 1