# Find general solution of the ODE \(\frac{d^2y}{dx^2} + 9y = 0\)

**1. Write down the given differential equation:**

\[

\frac{d^2y}{dx^2} + 9y = 0

\]

**2. Identify the characteristic equation:**

The characteristic equation associated with this second-order ODE is obtained by substituting \(y = e^{rx}\) into the equation:

\[

r^2 e^{rx} + 9 e^{rx} = 0

\]

**3. Simplify the characteristic equation:**

Factor out \(e^{rx}\):

\[

e^{rx} (r^2 + 9) = 0

\]

**4. Solve the characteristic equation:**

Set the equation equal to zero:

\[

r^2 + 9 = 0

\]

Solve for \(r\):

\[

r^2 = -9

\]

\[

r = \pm 3i

\]

Here, \(i\) is the imaginary unit, \(i = \sqrt{-1}\).

**5. Write down the solutions for \(r\):**

The solutions to the characteristic equation give us the roots \(r_1 = 3i\) and \(r_2 = -3i\).

**6. Write down the general solution:**

Since the roots are complex conjugates, the general solution to the ODE is:

\[

y(x) = C_1 e^{3ix} + C_2 e^{-3ix}

\]

**7. Express the solution in terms of real functions:**

Using Euler’s formula \(e^{ix} = \cos(x) + i\sin(x)\), rewrite the general solution:

\[

y(x) = C_1 (\cos(3x) + i\sin(3x)) + C_2 (\cos(3x) – i\sin(3x))

\]

\[

y(x) = (C_1 + C_2) \cos(3x) + i(C_1 – C_2) \sin(3x)

\]

**8. Combine constants to form a general real solution:**

Let \(A = C_1 + C_2\) and \(B = i(C_1 – C_2)\) where \(A\) and \(B\) are real constants:

\[

y(x) = A \cos(3x) + B \sin(3x)

\]

**Final Answer**

The general solution to the differential equation \(\frac{d^2y}{dx^2} + 9y = 0\) is:

### \[

y(x) = A \cos(3x) + B \sin(3x)

\]

where \(A\) and \(B\) are arbitrary constants.

This solution represents a linear combination of sine and cosine functions with a frequency determined by the square root of the coefficient of \(y\) in the differential equation.