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

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.