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  • Solve. x dy/dx + y + x sin (y/x) = 0

Solve.

    Q8.    x\frac{dy}{dx} - y + x\sin\left(\frac{y}{x}\right ) = 0

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\frac{dy}{dx}=\frac{y-x \sin(y/x)}{x} = F(x,y)...............................(i)

F(\mu x, \mu y)=\frac{\mu y-\mu x \sin(\mu y/\mu x)}{\mu x} = \mu^{0}.F(x,y)
it is a homogeneous equation

Now, to solve substitute y = vx

Differentiating on both sides wrt x
\frac{dy}{dx}= v +x\frac{dv}{dx}
                                 
Substitute this value in equation (i)

v+x\frac{dv}{dx}= v- \sin v = -\sin v
                                              \Rightarrow -\frac{dv}{\sin v} = -(cosec\ v)dv=\frac{dx}{x}

On integrating both sides we get;

\\\Rightarrow \log \left | cosec\ v-\cot v \right |=-\log x+ \log C\\ \Rightarrow cosec (y/x) - \cot (y/x) = C/x

= x[1-\cos (y/x)] = C \sin (y/x) Required solution

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