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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.

Solve the IVP: d^2y/dt^2 + 9y = cos(2t), with initial condition y(0) = 1 using Laplace transforms.

Learn to solve the second-order differential equation d²y/dt² + 9y = cos(2t) using Laplace transforms. This tutorial includes initial conditions, the Laplace and inverse Laplace transformations, partial fraction decomposition, and trigonometric function solutions.

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