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  • Solve the differential equation : <img alt="x\sin \left ( \frac{y}{x} \right )\frac{dy}{dx}+x=y\sin \left ( \frac{y}{x} \right )=0" src="https://learn.careers360.com/latex-image/?x%5Csin%20%5Cleft%20%28%20%5Cfrac%7By%7D%7Bx%7D%20%5Cright%20%29%5Cfrac%7

Solve the differential equation : x\sin \left ( \frac{y}{x} \right )\frac{dy}{dx}+x=y\sin \left ( \frac{y}{x} \right )=0

Given that x = 1 wheny=\frac{\pi}{2}

 

 
 
 
 
 

Answers (1)

Given: 

x\sin \left ( \frac{y}{x} \right )\frac{dy}{dx}+x=y\sin \left ( \frac{y}{x} \right )=0

and x = 1 wheny=\frac{\pi}{2}

\frac{d y}{d x}=\frac{y \sin \left(\frac{y}{x}\right)-x}{x \sin \left(\frac{y}{x}\right)}

\\ $Let $ v = \frac{y}{x} \\ y=v x \\\\ \frac{d y}{d x}=v+x \frac{d v}{d x} \\

\frac{d y}{d x}=\frac{y \sin \left(\frac{y}{x}\right)-x}{x \sin \left(\frac{y}{x}\right)}

\\v+x \frac{d v}{d x}=\frac{v \sin v-1}{\sin v} \\ x \frac{d v}{d x}=-\frac{1}{\sin v}\\

\\ \int \sin v d v=-\int \frac{1}{x} d x\\\\ \cos v=\log |x|+C \\ \cos \left(\frac{y}{x}\right)=\log |x|+C\\

\\ \text { Given } x=1 \text { when } y=\frac{\pi}{2} \\ \cos (\frac{\pi}{2}) = \log 1 +C \\ C=0\\\\ $The required solution $ \\ \cos \left(\frac{y}{x}\right)=\log |x|

 

Posted by

Safeer PP

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