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

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.