Let's go through the solution step-by-step using the Laplace transform.

Step a: Taking the Laplace Transform

Given the differential equation:

$y'' + 3y' + 2y = -3e^{-t}$

with initial conditions $y(0) = 5$ and $y'(0) = 2$.

Using the properties of Laplace transforms:

• $\mathcal{L}\{y''\} = s^2 Y(s) - sy(0) - y'(0)$
• $\mathcal{L}\{y'\} = s Y(s) - y(0)$
• $\mathcal{L}\{y\} = Y(s)$

Substitute into the equation:

$s^2 Y(s) - 5s - 2 + 3(s Y(s) - 5) + 2Y(s) = -\frac{3}{s+1}$

Simplifying:

$(s^2 + 3s + 2)Y(s) - 5s - 2 - 15 = -\frac{3}{s+1}$

Combine terms:

$(s^2 + 3s + 2)Y(s) = -\frac{3}{s+1} + 5s + 17$

So,

$\mathcal{L}\{y(t)\} = Y(s) = \frac{5s + 17}{s^2 + 3s + 2} - \frac{3}{(s+1)(s+2)}$

Step b: Partial Fraction Decomposition

Now, we decompose $Y(s)$ into partial fractions. Assuming:

$Y(s) = \frac{A}{s+1} + \frac{B}{(s+1)^2} + \frac{C}{s+2}$

After finding values for $A$, $B$, and $C$, substitute them back.

Step c: Inverse Laplace Transform

Finally, take the inverse Laplace transform term by term to find $y(t)$.

Would you like a detailed solution for each partial fraction step or just the final answer?

Follow-up Questions

1. How do initial conditions affect the Laplace transform of differential equations?
2. What are the steps for finding partial fraction decomposition?
3. Why is the inverse Laplace transform crucial in solving differential equations?
4. What role do the poles of $Y(s)$ play in the final solution?
5. How can you verify the solution once you have $y(t)$?

Tip

Remember, the Laplace transform simplifies solving differential equations by converting them into algebraic equations, making it easier to solve for $Y(s)$ before returning to the time domain.