The given problem is an integral equation of the following form:

$y'(t) - \int_0^t e^{t - x} y''(x) \, dx = y(t) + H(t - 1)$ with the initial conditions: $y(0) = 0, \quad y'(0) = 0.$

Steps to solve the equation:

1. Recognize the Structure: This equation involves a convolution-type term, given by the integral involving $y''(x)$. This suggests that transforming the equation (e.g., using Laplace transforms) might simplify it.

2. Laplace Transform: Apply the Laplace transform to both sides of the equation. Let $\mathcal{L}\{y(t)\} = Y(s)$. Recall that the Laplace transform of a derivative $y'(t)$ is $sY(s) - y(0)$, and for the second derivative, $\mathcal{L}\{y''(t)\} = s^2 Y(s) - sy(0) - y'(0)$. Given the initial conditions $y(0) = 0$ and $y'(0) = 0$, the transforms simplify.

   • Left-hand side: $\mathcal{L}\left\{y'(t)\right\} - \mathcal{L}\left\{\int_0^t e^{t - x} y''(x) dx\right\}.$ The first term is straightforward, yielding $sY(s)$.

   The second term involves the convolution of $e^t$ and $y''(t)$, which simplifies using the convolution theorem of Laplace transforms: $\mathcal{L}\left\{\int_0^t e^{t - x} y''(x) dx\right\} = \frac{1}{s - 1} \cdot (s^2 Y(s)).$

3. Transform the Rest: The right-hand side of the equation involves $y(t) + H(t - 1)$. Taking the Laplace transform of the right-hand side: $\mathcal{L}\left\{y(t)\right\} = Y(s), \quad \mathcal{L}\left\{H(t - 1)\right\} = \frac{e^{-s}}{s}.$

4. Solve for $Y(s)$: After applying the Laplace transform to the entire equation, you can collect the terms and solve for $Y(s)$.

5. Inverse Laplace Transform: Once you find $Y(s)$, take the inverse Laplace transform to get $y(t)$.

Would you like me to continue with detailed calculations, including the Laplace transform steps and finding $y(t)$? Let me know if you have any questions!

Follow-up questions:

1. Would you like to see the convolution process explained in detail?
2. Do you want the detailed steps for finding $Y(s)$ after taking the Laplace transform?
3. Should I explain how the Laplace transform of $H(t-1)$ is derived?
4. Are you familiar with the inverse Laplace transform method, or should I provide more details on that?
5. Do you need help verifying the initial conditions $y(0) = 0$ and $y'(0) = 0$ in the final solution?

Tip:

When solving integral equations using Laplace transforms, always make sure to carefully handle convolution terms, as they can simplify complex integrals significantly.