Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$…
Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ______
Solution
$\begin{aligned} & y e^{-x}=\int\left(e^{-x}+4 e^{-x} \sin x\right) d x \\ & y e^{-x}=-e^{-x}-2\left(e^{-x} \sin x e^{-x} \cos x\right)+C \\ & y=-1-2(\sin x+\cos x)+c e^x \\ & \because y(\pi)=1 \Rightarrow c=0 \\ & y(\pi / 2)=-1-2=-3 \\ & \text { Ans }=10-3=7\end{aligned}$