This problem requires solving a second-order differential equation using the Laplace Transform. Let me walk you through the process:

Steps:

1. Apply the Laplace Transform: Take the Laplace Transform of both sides of the equation $y'' + 9y' + 20y = 224e^{3t}$. Use the initial conditions $y(0) = -5$ and $y'(0) = 54$ to simplify.

2. Use Laplace Transform Properties:

   • The Laplace Transform of $y''$ is $s^2Y(s) - sy(0) - y'(0)$.
   • The Laplace Transform of $y'$ is $sY(s) - y(0)$.
   • The Laplace Transform of $e^{3t}$ is $\frac{1}{s - 3}$.

3. Set Up the Equation in Terms of $Y(s)$: After applying the transforms, solve for $Y(s)$, which represents the Laplace Transform of $y(t)$.

4. Perform Partial Fraction Decomposition: Simplify $Y(s)$ into a sum of fractions that can be easily inverted back to the time domain.

5. Apply the Inverse Laplace Transform: Use Laplace tables or properties to find $y(t)$ in terms of $e$.

Would you like me to delve deeper into any step or solve for $y(t)$ explicitly?

Related Questions:

1. What is the role of the initial conditions in solving differential equations with Laplace Transforms?
2. How does the exponential forcing term $e^{3t}$ influence the solution?
3. Why is partial fraction decomposition crucial in this method?
4. Can this approach handle non-homogeneous equations with arbitrary forcing functions?
5. How do we verify the solution obtained through the Laplace Transform?

Tip: Memorizing the basic Laplace Transform pairs and properties simplifies solving these problems significantly!