Find the particular solution of the differential equation :
x\frac{dy}{dx}\sin \left ( \frac{y}{x} \right )+x-y\sin \left ( \frac{y}{x} \right )= 0,  given that y\left ( 1 \right )= \frac{\pi }{2}\cdot

 

 

 

 
 
 
 
 

Answers (1)

x\frac{dy}{dx}\, \sin \left ( \frac{y}{x} \right )+x-y\, \sin \left ( \frac{y}{x} \right )= 0
\Rightarrow \frac{dy}{dx}= \frac{y}{x}-\frac{1}{\sin \left ( \frac{y}{x} \right )}
put y= vx\Rightarrow \frac{dy}{dx}= v+\frac{xdv}{dx}
\therefore v+\frac{xdv}{dx}= v-\frac{1}{\sin v}
\Rightarrow -\int \sin vdv= \int \frac{dx}{x}
\Rightarrow\cos v= \log \left | x \right |+c
\Rightarrow\cos \frac{y}{x}= \log \left | x \right |+c
given that y\left ( 1 \right )= \frac{\pi }{2},\: \cos \frac{\pi }{2}= \log \left | 1 \right |+c
\Rightarrow c= 0
Hence the required solution is \cos \frac{y}{x}= \log \left | x \right |

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