1. **State the problem:** Solve the integral differential equation $$y''(x) - e^x \int_0^x e^{-t} y''(t) \, dt = y(x) + u_1(x)$$ with initial conditions $$y(0) = 0$$ and $$y'(0) = 0$$. 2. **Rewrite the integral term:** Define $$I(x) = \int_0^x e^{-t} y''(t) \, dt$$ so the equation becomes $$y''(x) - e^x I(x) = y(x) + u_1(x).$$ 3. **Differentiate $I(x)$:** By the Fundamental Theorem of Calculus, $$I'(x) = e^{-x} y''(x).$$ 4. **Express $y''(x)$ from $I'(x)$:** $$y''(x) = e^x I'(x).$$ 5. **Substitute $y''(x)$ back into the original equation:** $$e^x I'(x) - e^x I(x) = y(x) + u_1(x).$$ Divide both sides by $e^x$: $$\cancel{e^x} I'(x) - \cancel{e^x} I(x) = \frac{y(x) + u_1(x)}{\cancel{e^x}}$$ which simplifies to $$I'(x) - I(x) = e^{-x} y(x) + e^{-x} u_1(x).$$ 6. **Recall $I(x)$ definition:** $$I(x) = \int_0^x e^{-t} y''(t) \, dt.$$ 7. **Relate $I(x)$ to $y'(x)$:** Integrate by parts or note that $$I(x) = e^{-x} y'(x) - y'(0) - \int_0^x (-e^{-t}) y'(t) \, dt = e^{-x} y'(x)$$ since $y'(0) = 0$ and the integral term cancels by differentiation. 8. **Use $I(x) = e^{-x} y'(x)$ in the differential equation:** $$I'(x) = -e^{-x} y'(x) + e^{-x} y''(x) = -I(x) + e^{-x} y''(x).$$ From step 5, $$I'(x) - I(x) = e^{-x} y(x) + e^{-x} u_1(x).$$ Substitute $I'(x)$: $$-I(x) + e^{-x} y''(x) - I(x) = e^{-x} y(x) + e^{-x} u_1(x)$$ which simplifies to $$e^{-x} y''(x) - 2 I(x) = e^{-x} y(x) + e^{-x} u_1(x).$$ Multiply both sides by $e^x$: $$y''(x) - 2 e^x I(x) = y(x) + u_1(x).$$ Recall from step 2 original equation: $$y''(x) - e^x I(x) = y(x) + u_1(x).$$ This implies $$y''(x) - 2 e^x I(x) = y''(x) - e^x I(x)$$ which means $$e^x I(x) = 0 \implies I(x) = 0.$$ 9. **Since $I(x) = 0$, from step 7:** $$e^{-x} y'(x) = 0 \implies y'(x) = 0.$$ 10. **Integrate $y'(x) = 0$:** $$y(x) = C,$$ and from initial condition $y(0) = 0$, we get $$C = 0,$$ so $$y(x) = 0.$$ 11. **Check the solution:** Substitute $y(x) = 0$ into the original equation: $$y''(x) - e^x \int_0^x e^{-t} y''(t) dt = 0 - e^x \cdot 0 = 0,$$ and right side $$y(x) + u_1(x) = 0 + u_1(x) = u_1(x).$$ For equality, $u_1(x)$ must be zero or the problem requires $u_1(x) = 0$ for this solution. **Final answer:** $$\boxed{y(x) = 0}$$ if $u_1(x) = 0$. If $u_1(x) \neq 0$, the problem requires further information or methods to solve.