Solve the differential equation :

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

 

 

 

 
 
 
 
 

Answers (1)

x\frac{\mathrm{d} y}{\mathrm{d} x}=y-x\tan \left ( \frac{y}{x} \right )   (Given) 

Let  y=vx

Then, \frac{\mathrm{d} y}{\mathrm{d} x}=v+x\frac{\mathrm{d} v}{\mathrm{d} x}

\therefore \frac{\mathrm x{d}y }{\mathrm{d} x}=y-x\tan \left ( \frac{y}{x} \right )

\Rightarrow x(v+\frac{\mathrm{ xd}v }{\mathrm{d} x})=vx-x\tan\left ( \frac{y}{x} \right )

\Rightarrow x(v+\frac{\mathrm{ xd}v }{\mathrm{d} x})=x(v-\tan v)

\Rightarrow xv+x^2\frac{\mathrm{d} v}{\mathrm{d} x}=xv-x\tan v

\Rightarrow x^2\frac{\mathrm{d} v}{\mathrm{d} x}=-x\tan v \Rightarrow \frac{\mathrm{xd}v }{\mathrm{d} x}=-\tan v

\Rightarrow\frac{\mathrm{d} v}{\mathrm{\tan} x}=\frac{\mathrm{-d}x }{\mathrm{} x} Then \int \cot v\: dv=-\int \frac{\mathrm{d} x}{\mathrm{} x}

\log \sin v=-\log x+\log c

\Rightarrow \log \sin \frac{y}{x}=\log \frac{c}{x}\Rightarrow {\sin \frac{y}{x}}=\frac{c}{x}

\Rightarrow x {\sin \frac{y}{x} }=c

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