# Solve the second-order ODE: $$y” – 4y = 0$$

1. Write down the given differential equation:

### $y” – 4y = 0$

2. Identify the characteristic equation:

The characteristic equation associated with this second-order ODE is obtained by substituting $$y = e^{rx}$$ into the equation:
$r^2 e^{rx} – 4 e^{rx} = 0$

3. Simplify the characteristic equation:

Factor out $$e^{rx}$$:
$e^{rx} (r^2 – 4) = 0$

4. Solve the characteristic equation:

Set the equation equal to zero:
$r^2 – 4 = 0$
Solve for $$r$$:
$r^2 = 4$
$r = \pm 2$

5. Write down the solutions for $$r$$:

The solutions to the characteristic equation give us the roots $$r_1 = 2$$ and $$r_2 = -2$$.

6. Write down the general solution:

Since the roots are distinct real numbers, the general solution to the ODE is:
$y(x) = C_1 e^{2x} + C_2 e^{-2x}$
where $$C_1$$ and $$C_2$$ are arbitrary constants to be determined by initial conditions.

The general solution to the differential equation $$y” – 4y = 0$$ is:
### $y(x) = C_1 e^{2x} + C_2 e^{-2x}$
where $$C_1$$ and $$C_2$$ are arbitrary constants.