We are tasked with solving the following initial value problem (IVP) using Laplace transforms:

$\frac{d^2 y}{dt^2} + 9y = \cos(2t), \quad y(0) = 1, \quad y'(0) = 0$

Step 1: Take the Laplace transform of both sides

Using the Laplace transform properties:

$\mathcal{L}\left\{\frac{d^2 y}{dt^2}\right\} = s^2 Y(s) - s y(0) - y'(0)$ Substituting the initial conditions $y(0) = 1$ and $y'(0) = 0$:

$\mathcal{L}\left\{\frac{d^2 y}{dt^2}\right\} = s^2 Y(s) - s$

The Laplace transform of $9y$ is:

$\mathcal{L}\{9y\} = 9 Y(s)$

For $\cos(2t)$, the Laplace transform is:

$\mathcal{L}\{\cos(2t)\} = \frac{s}{s^2 + 4}$

Step 2: Formulate the transformed equation

Substituting all of these into the original equation:

$(s^2 Y(s) - s) + 9 Y(s) = \frac{s}{s^2 + 4}$

Step 3: Solve for $Y(s)$

Combine terms to solve for $Y(s)$:

$(s^2 + 9) Y(s) - s = \frac{s}{s^2 + 4}$

Move the $s$-term to the other side:

$(s^2 + 9) Y(s) = s + \frac{s}{s^2 + 4}$

Now solve for $Y(s)$:

$Y(s) = \frac{s}{s^2 + 9} + \frac{s}{(s^2 + 9)(s^2 + 4)}$

Step 4: Perform inverse Laplace transform

We now take the inverse Laplace transform of each term:

1. The inverse Laplace transform of $\frac{s}{s^2 + 9}$ is $\cos(3t)$.
2. For the second term $\frac{s}{(s^2 + 9)(s^2 + 4)}$, we use partial fractions:

$\frac{s}{(s^2 + 9)(s^2 + 4)} = \frac{A s}{s^2 + 9} + \frac{B s}{s^2 + 4}$

Solving for $A$ and $B$, we obtain $A = \frac{1}{5}$ and $B = -\frac{1}{5}$.

Thus, the second term becomes:

$\frac{1}{5} \cos(3t) - \frac{1}{5} \cos(2t)$

Step 5: Final solution

The complete solution is:

$y(t) = \cos(3t) + \frac{1}{5} (\cos(3t) - \cos(2t))$

Simplifying further:

$y(t) = \frac{6}{5} \cos(3t) - \frac{1}{5} \cos(2t)$

This is the solution to the IVP.

Would you like more details or have any questions on the solution process?

Here are five related questions to consider:

1. How do we compute the Laplace transform of higher-order derivatives?
2. What is the general procedure for solving second-order differential equations using Laplace transforms?
3. How do initial conditions affect the Laplace-transformed equation?
4. Can you derive the Laplace transform of other trigonometric functions, such as sine or tangent?
5. How does partial fraction decomposition assist in inverse Laplace transforms?

Tip: When solving differential equations using Laplace transforms, always remember to carefully handle the initial conditions during the transformation process.