Mathematics - Differential Equation Question with Solution | TestHub

According to the condition, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{x}-{\cos }^2\frac{y}{x}\dots \text{ (i) }$ This is a homogeneous differential equation Substituting $y=vx$, we get $\begin{array}{l}v+x\frac{\mathrm{dv}}{\mathrm{dx}}=v-{\cos }^{2}v\\ \Rightarrow x\frac{\mathrm{dv}}{\mathrm{dx}}=-{\cos }^{2}v\\ \Rightarrow \int {\sec }^2\mathrm{vdv}=-\int \frac{\mathrm{dx}}{\mathrm{x}}\\ \Rightarrow \tan v=-\log x+\mathrm{C}\\ \Rightarrow \tan \frac{y}{x}+\log x=\mathrm{C}\end{array}$ Substituting $\mathrm{x}=1,y=\frac{\pi }{4}$, we get $\mathrm{C}=1$ Thus, we get $\tan \left(\frac{y}{x}\right)+\log x=1$ which is the required solution.