To solve this problem, let's go through each part step-by-step using the Laplace transform.

Problem Breakdown:

Given the initial value problem: $y'' - 3y' + 2y = -7e^t, \quad y(0) = 4, \quad y'(0) = -8.$

Step-by-Step Solution

Part (a): Take the Laplace Transform of Both Sides

1. Apply the Laplace transform to the differential equation. Recall that: $\mathcal{L}\{y''\}(s) = s^2 Y(s) - s y(0) - y'(0),$ $\mathcal{L}\{y'\}(s) = s Y(s) - y(0),$ $\mathcal{L}\{y\}(s) = Y(s).$

2. Substitute the initial conditions $y(0) = 4$ and $y'(0) = -8$ into the transformed equation.

Thus, taking the Laplace transform of both sides:

$s^2 Y(s) - s \cdot 4 - (-8) - 3(s Y(s) - 4) + 2 Y(s) = \frac{-7}{s - 1}.$

3. Simplify the equation: $(s^2 - 3s + 2)Y(s) - 4s + 8 + 12 = \frac{-7}{s - 1}.$

   Simplify further to get $\mathcal{L}\{y(t)\}(s)$ in terms of $Y(s)$.

Part (b): Partial Fraction Decomposition

Rewrite $\mathcal{L}\{y(t)\}(s)$ in a form suitable for partial fraction decomposition: $\mathcal{L}\{y(t)\}(s) = \frac{A}{s - a} + \frac{B}{(s - a)^2} + \frac{C}{s - b}.$

Solve for constants $A$, $B$, and $C$ by equating terms.

Part (c): Take the Inverse Laplace Transform

Once the partial fractions are decomposed, apply the inverse Laplace transform term-by-term to obtain $y(t)$.

Would you like a detailed expansion of each step, particularly in solving for the partial fraction constants, or do you have any specific questions about this process?

Related Questions to Expand Understanding

1. How do initial conditions influence the Laplace transform solution?
2. What are the common forms for partial fraction decomposition in Laplace transforms?
3. How do we handle non-homogeneous terms like $-7e^t$ in Laplace transforms?
4. What are the steps to perform inverse Laplace transformations after decomposition?
5. How do we use properties of Laplace transforms to simplify expressions?

Tip:

When solving differential equations with Laplace transforms, always simplify each term after transforming to avoid complex partial fraction decompositions later on.