Find the general solution of the following reducible​ second-order differential equation. Assume​ x, y​ and/or y prime positive where helpful.
y prime prime plus 64 y equals 0


The given second-order differential equation is:

\[
y'' + 64y = 0
\]

This is a linear homogeneous second-order differential equation with constant coefficients. To solve it, we proceed as follows:

### Step 1: Find the characteristic equation
The corresponding characteristic equation is obtained by assuming a solution of the form \( y = e^{rt} \), where \( r \) is a constant. Substituting \( y = e^{rt} \) into the differential equation:

\[
r^2 e^{rt} + 64 e^{rt} = 0
\]

Dividing by \( e^{rt} \) (which is non-zero):

\[
r^2 + 64 = 0
\]

### Step 2: Solve the characteristic equation
Solve the characteristic equation \( r^2 + 64 = 0 \):

\[
r^2 = -64
\]

\[
r = \pm 8i
\]

This gives two complex roots: \( r = 8i \) and \( r = -8i \).

### Step 3: General solution for complex roots
For complex roots of the form \( r = \pm \alpha i \), the general solution to the differential equation is:

\[
y(t) = c_1 \cos(\alpha t) + c_2 \sin(\alpha t)
\]

In this case, \( \alpha = 8 \), so the general solution is:

\[
y(t) = c_1 \cos(8t) + c_2 \sin(8t)
\]

### Final Answer:
The general solution to the differential equation \( y'' + 64y = 0 \) is:

\[
y(t) = c_1 \cos(8t) + c_2 \sin(8t)
\]

where \( c_1 \) and \( c_2 \) are arbitrary constants.

Would you like further details or clarifications?

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Here are 5 related questions to deepen your understanding:
1. How do we handle characteristic equations with real roots?
2. What happens if the characteristic equation has repeated roots?
3. How would the general solution change if the equation had a non-homogeneous term?
4. Can we solve for \( c_1 \) and \( c_2 \) if initial conditions are provided?
5. What is the significance of the complex roots in this differential equation?

**Tip:** For second-order linear homogeneous differential equations with constant coefficients, always start by solving the characteristic equation.

Find the general solution of the following reducible second-order differential equation. Assume x, y and/or y prime positive where helpful. y prime prime plus 64 y equals 0

Discover the step-by-step process to find the general solution of the second-order differential equation y'' + 64y = 0. This guide covers characteristic equations, complex roots, and general solutions, ideal for undergraduate math students.

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