$$ \ddot { x } + \omega ^ { 2 } x = 0 $$

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The given equation is a second-order linear homogeneous differential equation, which is commonly known as the simple harmonic oscillator equation. The general form of the equation is:

$$
\ddot{x} + \omega^2 x = 0
$$

where $$ \ddot{x} $$ is the second derivative of $$ x $$ with respect to time, and $$ \omega $$ is a constant representing the angular frequency.

### Step 1: Characteristic Equation
To solve this differential equation, we first assume a solution of the form:

$$
x(t) = e^{rt}
$$

where $$ r $$ is a constant. Substituting this into the differential equation gives:

$$
r^2 e^{rt} + \omega^2 e^{rt} = 0
$$

Factoring out $$ e^{rt} $$ (which is never zero), we obtain the characteristic equation:

$$
r^2 + \omega^2 = 0
$$

### Step 2: Solve the Characteristic Equation
Now, we solve for $$ r $$:

$$
r^2 = -\omega^2
$$
$$
r = \pm i\omega
$$

### Step 3: General Solution
The general solution of the differential equation can be expressed in terms of sine and cosine functions:

$$
x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)
$$

where $$ C_1 $$ and $$ C_2 $$ are constants determined by initial conditions.

### Summary
The solution to the differential equation $$ \ddot{x} + \omega^2 x = 0 $$ is:

$$
x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)
$$

This represents the motion of a simple harmonic oscillator, where $$ C_1 $$ and $$ C_2 $$ can be determined based on the initial position and velocity of the system.
The solution to the equation $$ \ddot{x} + \omega^2 x = 0 $$ is: $$ x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t) $$ where $$ C_1 $$ and $$ C_2 $$ are constants determined by initial conditions.

\( \ddot { x } + \omega ^ { 2 } x = 0 \)

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