$$\begin{aligned} & \frac{d y}{d x}-y=1+4 \sin x \\ & \text { Integrating factor }=e^{-\int d x}=e^{-x} \end{aligned}$$

Solution is $$y e^{-x}=\int(1+4 \sin x) e^{-x} d x$$

$$\begin{aligned} & =-e^{-x}+2 \cdot e^{-x}(-\sin x-\cos x)+C \\ y(\pi) & =1 \Rightarrow C=0 \end{aligned}$$

Hence $$y(x)=-1-2(\sin x+\cos x)$$

$$y\left(\frac{\pi}{2}\right)+10=7$$