Discussion Forum

The solution of the differential equation $\frac{dy}{dx}=\tan\left(\frac{y}{x}\right)+\frac{y}{x}$  is 

Answer:

Explanation:

 Given, 

 $\frac{dy}{dx}=\tan\left(\frac{y}{x}\right)+\frac{y}{x}$ ..........................(i)

 Clearly , the given different equation is homogeneous

 On putting y=Vx

$\frac{dy}{dx}=V+x\frac{dV}{dx}$     in Eq.(i) we get

 $V+x\frac{dV}{dx}=\tan V+V$

 $\Rightarrow$    $x\frac{dV}{dx}=\tan V$

 $\Rightarrow$      $\frac{1}{\tan V}dV=\frac{1}{x}dx$

On integrating both sides, we get

$\int \frac{1}{\tan V}dV=\int\frac{1}{x}dx$

$\Rightarrow$   $\int \cot VdV=\log x+\log c$

$\Rightarrow$     log sin x = log x+ log c

$\Rightarrow$   log sin V= log(xc)

$\Rightarrow$   sin v=xc

$\therefore$     $\sin (\frac{y}{x})=xc$