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.
