Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& \frac{dy}{dx}+2y\tan x=\sin x\\ & \text{ I.F. }=e^{2\int \tan xdx}={\sec }^{2}x\\ & y{\sec }^{2}x=\int \frac{\sin x}{{\cos }^{2}x}dx\\ & =\int \tan x\sec xdx\\ & =\sec x+C\\ & C=-2\end{array}$ $\begin{array}{rl}& y=\cos x-2{\cos }^{2}x\\ & y\left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}-1\\ & y^{\prime }=-\sin x+4\cos x\sin x\\ & y^{\prime }\left(\frac{\pi }{4}\right)=-\frac{1}{\sqrt{2}}+2\\ & y^{\prime }\left(\frac{\pi }{4}\right)+y\left(\frac{\pi }{4}\right)=1\end{array}$