Let the solution $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ satisfy $$y(\pi) = 1$$. Then $$y\left(\frac{\pi}{2}\right) + 10$$ is equal to ______.

We need to solve the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ with the condition $$y(\pi) = 1$$ and then find $$y\left(\frac{\pi}{2}\right) + 10$$.

This is a first-order linear ODE of the form $$\frac{dy}{dx} + P(x)y = Q(x)$$ where $$P(x) = -1$$ and $$Q(x) = 1 + 4\sin x$$. We use an integrating factor to solve it.

The integrating factor is given by $$ \text{IF} = e^{\int P(x)\,dx} = e^{\int -1\,dx} = e^{-x} $$

Multiplying both sides of the original equation by this factor and applying the product rule yields $$ y\,e^{-x} = \int (1 + 4\sin x)\,e^{-x}\,dx + C $$

The integral of the first term is $$\int e^{-x}\,dx = -e^{-x}\,. $$

To evaluate the second integral $$\int 4\sin x\,e^{-x}\,dx$$ we use the standard result $$\int e^{ax}\sin(bx)\,dx = \frac{e^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2}$$ with $$a = -1$$ and $$b = 1$$. Hence $$ \int e^{-x}\sin x\,dx = \frac{e^{-x}(-\sin x - \cos x)}{(-1)^2 + 1^2} = \frac{e^{-x}(-\sin x - \cos x)}{2} $$ and therefore $$ \int 4\sin x\,e^{-x}\,dx = 4 \cdot \frac{e^{-x}(-\sin x - \cos x)}{2} = -2e^{-x}(\sin x + \cos x)\,. $$

Combining these results gives $$ y\,e^{-x} = -e^{-x} - 2e^{-x}(\sin x + \cos x) + C\,. $$

Multiplying through by $$e^{x}$$ to solve for $$y$$ leads to $$ y = -1 - 2(\sin x + \cos x) + Ce^{x}\,. $$

Applying the initial condition $$y(\pi) = 1$$ gives the equation $$ 1 = -1 - 2(\sin\pi + \cos\pi) + Ce^{\pi}\,. $$ Since $$\sin\pi = 0$$ and $$\cos\pi = -1$$ this becomes $$1 = -1 - 2(0 + (-1)) + Ce^{\pi} = 1 + Ce^{\pi}\,, $$ so $$Ce^{\pi} = 0$$ and hence $$C = 0$$.

Substituting back yields the particular solution $$ y = -1 - 2(\sin x + \cos x)\,. $$

Evaluating at $$x = \frac{\pi}{2}$$ gives $$ y\left(\frac{\pi}{2}\right) = -1 - 2\Bigl(\sin\frac{\pi}{2} + \cos\frac{\pi}{2}\Bigr) = -1 - 2(1 + 0) = -3\,. $$

Therefore, $$y\left(\frac{\pi}{2}\right) + 10 = -3 + 10 = 7\,. $$

The answer is 7.