Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& {\sec }^{2}y\frac{dy}{dx}+2x\sin y\text{ secy }=x^3\cos y\text{ secy }\\ & {\sec }^{2}y\frac{dy}{dx}+2x\tan y=x^3\\ & \mathrm{tany}=t\Rightarrow {\sec }^{2}y\frac{dy}{dx}=\frac{dt}{dx}\\ & \frac{dt}{dx}+2xt=x^3,\text{ If }=e^{\int 2xdx}=e^{x^2}\\ & t{x}^{x^2}=\int x^3\cdot e^{x^2}dx+c\\ & x^2=Z\Rightarrow t\cdot e^Z=\frac{1}{2}\int e^Z\cdot ZdZ=\frac{1}{2}\left[e^Z\cdot Z-e^Z\right]+c\\ & 2\tan y=\left(x^2-1\right)+2e^{-x^2}\\ & y(1)=0\Rightarrow c=0\Rightarrow y(\sqrt{3})=\frac{\pi }{4}\end{array}$