We are given the non-homogeneous second-order differential equation:

\[
y'' + 3y' = 108\sin(3t) + 36\cos(3t)
\]

### Step 1: Solve the homogeneous equation

The corresponding homogeneous equation is:

\[
y'' + 3y' = 0
\]

This can be rewritten as:

\[
r^2 + 3r = 0
\]

Factoring out \(r\), we get:

\[
r(r + 3) = 0
\]

Thus, the roots are \(r = 0\) and \(r = -3\). Therefore, the general solution to the homogeneous equation is:

\[
y_h(t) = C_1 + C_2 e^{-3t}
\]

### Step 2: Solve the particular solution

For the non-homogeneous part \(108\sin(3t) + 36\cos(3t)\), we propose a particular solution of the form:

\[
y_p(t) = A\sin(3t) + B\cos(3t)
\]

We now substitute \(y_p(t)\) into the original equation. First, we calculate the derivatives of \(y_p(t)\):

\[
y_p'(t) = 3A\cos(3t) - 3B\sin(3t)
\]

\[
y_p''(t) = -9A\sin(3t) - 9B\cos(3t)
\]

Now substitute \(y_p(t)\), \(y_p'(t)\), and \(y_p''(t)\) into the original equation \(y'' + 3y' = 108\sin(3t) + 36\cos(3t)\):

\[
(-9A\sin(3t) - 9B\cos(3t)) + 3(3A\cos(3t) - 3B\sin(3t)) = 108\sin(3t) + 36\cos(3t)
\]

Simplify both sides:

\[
(-9A\sin(3t) - 9B\cos(3t)) + (9A\cos(3t) - 9B\sin(3t)) = 108\sin(3t) + 36\cos(3t)
\]

Now group the terms involving \(\sin(3t)\) and \(\cos(3t)\):

\[
(-9A - 9B)\sin(3t) + (9A - 9B)\cos(3t) = 108\sin(3t) + 36\cos(3t)
\]

Equating the coefficients of \(\sin(3t)\) and \(\cos(3t)\):

\[
-9A + 9B = 108 \quad \text{(for } \sin(3t) \text{)}
\]
\[
9A - 9B = 36 \quad \text{(for } \cos(3t) \text{)}
\]

### Step 3: Solve for \(A\) and \(B\)

From the first equation:

\[
-9A - 9B = 108 \quad \Rightarrow \quad A + B = -12
\]

From the second equation:

\[
9A - 9B = 36 \quad \Rightarrow \quad A - B = 4
\]

Now solve this system of equations. Add the two equations:

\[
(A + B) + (A - B) = -12 + 4
\]
\[
2A = -8 \quad \Rightarrow \quad A = -4
\]

Substitute \(A = -4\) into \(A + B = -12\):

\[
-4 + B = -12 \quad \Rightarrow \quad B = -8
\]

### Step 4: General solution

The particular solution is:

\[
y_p(t) = -4\sin(3t) - 8\cos(3t)
\]

Thus, the general solution to the differential equation is the sum of the homogeneous and particular solutions:

\[
y(t) = y_h(t) + y_p(t) = C_1 + C_2 e^{-3t} - 4\sin(3t) - 8\cos(3t)
\]

Would you like more details on any step or further explanations?

Here are some related questions for further practice:
1. How would you solve if the right-hand side had \(e^{3t}\) instead of trigonometric terms?
2. What happens if the homogeneous equation has complex roots?
3. How can this method be applied to higher-order differential equations?
4. What are other methods for solving non-homogeneous differential equations?
5. How does the solution change if we modify the coefficients of sine and cosine terms?

**Tip:** When dealing with sinusoidal terms, always check if a particular solution of similar form fits the equation before solving for constants!

This detailed solution walks through solving the non-homogeneous differential equation y'' + 3y' = 108sin(3t) + 36cos(3t). The problem covers key topics such as the homogeneous equation, the method of undetermined coefficients, and superposition principle. Ideal for students studying calculus or differential equations.

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