To solve this second-order linear non-homogeneous differential equation with given initial conditions:

$\frac{d^2y}{dt^2} - \frac{dy}{dt} - 2y = 1 - t^2 + e^t - \cos t$ with initial conditions: $y(0) = 1, \quad \frac{dy}{dt}\Big|_{t=0} = 0,$

we'll break down the process as follows:

1. Solve the corresponding homogeneous equation: $\frac{d^2y}{dt^2} - \frac{dy}{dt} - 2y = 0.$ This will give the complementary solution, $y_c$.

2. Find a particular solution $y_p$ for the non-homogeneous part, $1 - t^2 + e^t - \cos t$, using an appropriate method like undetermined coefficients or variation of parameters.

3. Combine the complementary and particular solutions to get the general solution: $y(t) = y_c + y_p.$

4. Apply the initial conditions $y(0) = 1$ and $\frac{dy}{dt}\Big|_{t=0} = 0$ to determine any constants in the general solution.

Would you like a detailed, step-by-step solution for each part of this process?

Here are five related questions to delve deeper:

1. How do you find the complementary solution for a second-order differential equation?
2. What methods are effective for finding particular solutions for non-homogeneous differential equations?
3. Why are initial conditions crucial for determining the specific solution?
4. How would the solution change if the non-homogeneous part were different?
5. Can you use variation of parameters for this type of problem?

Tip: Remember that finding a particular solution often requires testing multiple forms when the non-homogeneous term has various components (like polynomials, exponentials, and trigonometric functions).