The general solution of the differential equation. $\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d…

The general solution of the differential equation. $\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right] d y=0 \text { is }$
  1. $\mathrm{y}^2 \sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\mathrm{k}$
  2. $\mathrm{x} \sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\mathrm{k}$
  3. $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\mathrm{k}$
  4. $y \sin \left(\frac{y}{x}\right)=k$

Solution

We have $\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right] d y=0$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right)}{\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)}$ Put $\frac{y}{x}=v \Rightarrow y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$ $\begin{aligned} & \therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}}{\frac{1}{\mathrm{v}} \sin \mathrm{v}+\cos \mathrm{v}}=\frac{\mathrm{v}^2 \cos \mathrm{v}}{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}} \Rightarrow \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{v} \sin \mathrm{v}}{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}} \\ & \therefore \int \frac{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}} \mathrm{dv}=\int \frac{-\mathrm{dx}}{\mathrm{x}} \Rightarrow \int \frac{1}{\mathrm{v}} \mathrm{dv}+\int \cot \mathrm{v} d \mathrm{v}=-\int \frac{\mathrm{dx}}{\mathrm{x}} \\ & \therefore \log |\mathrm{v}|+\log |+\sin \mathrm{x}|=-\log |\mathrm{x}|+\log \mathrm{k} \Rightarrow \log |(\mathrm{v})(\sin \mathrm{v})(\mathrm{x})| \\ & =\log \mathrm{k} \end{aligned}$ $\therefore y \sin \left(\frac{y}{x}\right)=k$

Asked in: MHT CET 2021 (21 Sep Shift 2)

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