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The given differential equation is:

\[
y'' + 8y' = 512 \sin(8t) + 1152 \cos(8t)
\]

with initial conditions \( y(0) = 4 \) and \( y'(0) = 5 \).

### Step 1: Solve the homogeneous equation

The homogeneous part of the equation is:

\[
y'' + 8y' = 0
\]

The characteristic equation for this is:

\[
r^2 + 8r = 0
\]

Factoring:

\[
r(r + 8) = 0
\]

So, \( r = 0 \) or \( r = -8 \).

Thus, the general solution to the homogeneous equation is:

\[
y_h(t) = C_1 + C_2 e^{-8t}
\]

### Step 2: Solve the particular solution

For the non-homogeneous part \( 512 \sin(8t) + 1152 \cos(8t) \), we try a particular solution of the form:

\[
y_p(t) = A \sin(8t) + B \cos(8t)
\]

Now, we find the first and second derivatives of \( y_p(t) \):

\[
y_p'(t) = 8A \cos(8t) - 8B \sin(8t)
\]
\[
y_p''(t) = -64A \sin(8t) - 64B \cos(8t)
\]

Substituting \( y_p(t) \), \( y_p'(t) \), and \( y_p''(t) \) into the original equation:

\[
(-64A \sin(8t) - 64B \cos(8t)) + 8(8A \cos(8t) - 8B \sin(8t)) = 512 \sin(8t) + 1152 \cos(8t)
\]

Simplifying:

\[
(-64A \sin(8t) - 64B \cos(8t)) + (64A \cos(8t) - 64B \sin(8t)) = 512 \sin(8t) + 1152 \cos(8t)
\]

Group terms:

\[
(-64A - 64B) \sin(8t) + (64A - 64B) \cos(8t) = 512 \sin(8t) + 1152 \cos(8t)
\]

Now, equate coefficients of \( \sin(8t) \) and \( \cos(8t) \):

\[
-64B = 512 \quad \Rightarrow \quad B = -8
\]
\[
64A = 1152 \quad \Rightarrow \quad A = 18
\]

Thus, the particular solution is:

\[
y_p(t) = 18 \sin(8t) - 8 \cos(8t)
\]

### Step 3: General solution

The general solution is the sum of the homogeneous and particular solutions:

\[
y(t) = C_1 + C_2 e^{-8t} + 18 \sin(8t) - 8 \cos(8t)
\]

### Step 4: Apply initial conditions

Using the initial condition \( y(0) = 4 \):

\[
C_1 + C_2 e^0 + 18 \sin(0) - 8 \cos(0) = 4
\]
\[
C_1 + C_2 - 8 = 4 \quad \Rightarrow \quad C_1 + C_2 = 12
\]

Using the second initial condition \( y'(0) = 5 \):

\[
y'(t) = -8C_2 e^{-8t} + 18 \cdot 8 \cos(8t) + 8 \cdot 8 \sin(8t)
\]

At \( t = 0 \):

\[
y'(0) = -8C_2 + 144 = 5
\]
\[
-8C_2 + 144 = 5 \quad \Rightarrow \quad -8C_2 = -139 \quad \Rightarrow \quad C_2 = \frac{139}{8}
\]

Substitute \( C_2 \) into \( C_1 + C_2 = 12 \):

\[
C_1 + \frac{139}{8} = 12 \quad \Rightarrow \quad C_1 = 12 - \frac{139}{8} = \frac{96}{8} - \frac{139}{8} = \frac{-43}{8}
\]

### Final solution:

\[
y(t) = \frac{-43}{8} + \frac{139}{8} e^{-8t} + 18 \sin(8t) - 8 \cos(8t)
\]

This is the solution to the given differential equation.

Do you have any questions or would you like more details on any step? 

Here are 5 questions to consider:
1. How does the method of undetermined coefficients work?
2. What other methods can solve non-homogeneous differential equations?
3. Can this technique apply to different forcing functions like \( e^{at} \)?
4. What happens when there is resonance in a non-homogeneous equation?
5. How would the solution change if the initial conditions were different?

**Tip**: Always verify the particular solution by substituting it back into the original equation.

Find the solution of y'' + 8y' = 512 sin(8t) + 1152 cos(8t) with y(0) = 4 and y'(0) = 5.

Explore the step-by-step solution of the differential equation y'' + 8y' = 512 sin(8t) + 1152 cos(8t) with given initial conditions y(0) = 4 and y'(0) = 5. Learn about the method of undetermined coefficients, solving homogeneous equations, and applying initial conditions for the final solution. Suitable for Grade 12 and college-level students.

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