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The given differential equation is:

\[
y'' + 9y' = 648 \sin(9t) + 1458 \cos(9t)
\]

with the initial conditions \( y(0) = 5 \) and \( y'(0) = 1 \).

### Solution Strategy:
This is a non-homogeneous second-order linear differential equation. To solve it, we will follow these steps:

1. **Solve the homogeneous equation**:
   \[
   y'' + 9y' = 0
   \]
   
2. **Find a particular solution** to the non-homogeneous equation.

3. **Combine the homogeneous and particular solutions** to form the general solution.

4. **Use the initial conditions** to determine the constants in the solution.

### Step 1: Solve the homogeneous equation
The homogeneous equation is:
\[
y'' + 9y' = 0
\]

To solve, assume \( y_h = e^{rt} \), and substitute into the homogeneous equation:
\[
r^2 e^{rt} + 9r e^{rt} = 0
\]
Factor:
\[
e^{rt}(r^2 + 9r) = 0
\]
This gives the characteristic equation:
\[
r(r + 9) = 0
\]
Thus, \( r = 0 \) or \( r = -9 \).

So, the general solution to the homogeneous equation is:
\[
y_h(t) = C_1 + C_2 e^{-9t}
\]

### Step 2: Find the particular solution
The non-homogeneous term is \( 648 \sin(9t) + 1458 \cos(9t) \). Since the right-hand side is a combination of sine and cosine, we try a particular solution of the form:
\[
y_p(t) = A \sin(9t) + B \cos(9t)
\]

Now, calculate the first and second derivatives of \( y_p(t) \):
\[
y_p'(t) = 9A \cos(9t) - 9B \sin(9t)
\]
\[
y_p''(t) = -81A \sin(9t) - 81B \cos(9t)
\]

Substitute \( y_p(t) \), \( y_p'(t) \), and \( y_p''(t) \) into the original equation \( y'' + 9y' = 648 \sin(9t) + 1458 \cos(9t) \):
\[
(-81A \sin(9t) - 81B \cos(9t)) + 9(9A \cos(9t) - 9B \sin(9t)) = 648 \sin(9t) + 1458 \cos(9t)
\]
Simplify the left-hand side:
\[
-81A \sin(9t) - 81B \cos(9t) + 81A \cos(9t) - 81B \sin(9t) = 648 \sin(9t) + 1458 \cos(9t)
\]
Group the terms involving \( \sin(9t) \) and \( \cos(9t) \):
\[
(-81A - 81B) \sin(9t) + (81A - 81B) \cos(9t) = 648 \sin(9t) + 1458 \cos(9t)
\]
Equating the coefficients of \( \sin(9t) \) and \( \cos(9t) \), we get the system of equations:
\[
-81A - 81B = 648 \quad \text{(1)}
\]
\[
81A - 81B = 1458 \quad \text{(2)}
\]

From equation (1):
\[
A + B = -8
\]

From equation (2):
\[
A - B = 18
\]

Add the two equations:
\[
2A = 10 \quad \Rightarrow \quad A = 5
\]
Substitute \( A = 5 \) into \( A + B = -8 \):
\[
5 + B = -8 \quad \Rightarrow \quad B = -13
\]

Thus, the particular solution is:
\[
y_p(t) = 5 \sin(9t) - 13 \cos(9t)
\]

### Step 3: General solution
The general solution to the differential equation is the sum of the homogeneous and particular solutions:
\[
y(t) = C_1 + C_2 e^{-9t} + 5 \sin(9t) - 13 \cos(9t)
\]

### Step 4: Apply initial conditions
Now, use the initial conditions \( y(0) = 5 \) and \( y'(0) = 1 \).

1. \( y(0) = 5 \):
\[
C_1 + C_2 e^{0} + 5 \sin(0) - 13 \cos(0) = 5
\]
\[
C_1 - 13 = 5 \quad \Rightarrow \quad C_1 = 18
\]

2. \( y'(t) \):
\[
y'(t) = -9C_2 e^{-9t} + 45 \cos(9t) + 117 \sin(9t)
\]
At \( t = 0 \), \( y'(0) = 1 \):
\[
-9C_2 + 45 \cos(0) + 117 \sin(0) = 1
\]
\[
-9C_2

Find the solution of y'' + 9y' = 648 sin(9t) + 1458 cos(9t) with y(0) = 5 and y'(0) = 1.

Learn how to solve the non-homogeneous second-order differential equation y'' + 9y' = 648 sin(9t) + 1458 cos(9t) with initial conditions y(0) = 5 and y'(0) = 1. This detailed step-by-step solution covers methods like solving homogeneous equations and finding particular solutions.

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