Students now know how to represent harmonic motion and have discussed oscillations from an energy diagram perspective. It is time to set up Newton's law and obtain that same representation as a solution to a differential equation.

Clearly identify “the force”, \(F\left( x \right)=-m\omega _{0}^{2}x\) and set it equal to the acceleration times m, \(a=m\frac{d^{2}x}{dt^{2}}\). Students often think that \(m\frac{d^{2}x}{dt^{2}}\) is “the force”.

Solve the differential equation postulating a solution \(Ce^{pt}\), and obtain a solution in the “C-form” with \(e^{\pm i\omega _{0}t}\) terms. The system can be a simple harmonic oscillator or a plane pendulum approximated to SHM. Refer to initial condition examples to find the coefficients.