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# Solve for particular solution. dy/dx - y/x + cosec ( y/x) = 0; y = 0 when x = 1

Solve for particular solution.

Q14.    $\frac{dy}{dx} - \frac{y}{x} + \textup{cosec}\left (\frac{y}{x} \right ) = 0;\ y = 0 \ \textup{when}\ x = 1$

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

the above eq is homogeneous. So,
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)

$\\=v+x\frac{dv}{dx}=v- cosec\ v\\ =x\frac{dv}{dx} = -cosec\ v\\ =-\frac{dv}{cosec\ v}= \frac{dx}{x}\\ =-\sin v dv = \frac{dx}{x}$

on integrating both sides, we get;

$\\=cos\ v = \log x +\log C =\log Cx\\ =\cos(y/x)= \log Cx$.................................(ii)

now y = 0 and x =1 , we get

$C =e^{1}$

put the value of C in eq 2

$\cos(y/x)=\log \left | ex \right |$

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