The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\tan…

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\tan \left(\frac{\mathrm{y}}{\mathrm{x}}\right)+\frac{\mathrm{y}}{\mathrm{x}}$ is
  1. $\sin \left(\frac{y}{x}\right)=c y$
  2. $\cos \left(\frac{y}{x}\right)=c y$
  3. $\cos \left(\frac{y}{x}\right)=c x$
  4. $\sin \left(\frac{y}{x}\right)=c x$

Solution

$\begin{aligned} & \frac{d y}{d x}=\tan \left(\frac{y}{x}\right)+\frac{y}{x} \\ & \text { Put } \frac{y}{x}=v \Rightarrow y=x v \Rightarrow \frac{d y}{d x}=x \frac{d v}{d x}+v \\ & \therefore \quad v+x \frac{d v}{d x}=\tan v+v \Rightarrow x \frac{d v}{d x}=\tan v \\ & \therefore \quad \int \frac{d v}{\tan v}=\int \frac{d x}{x} \\ & \therefore \quad \log |\sin v|=\log |x|+\log |c| \Rightarrow \sin v=x c \\ & \therefore \quad \sin \left(\frac{y}{x}\right)=x c\end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 2)

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