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## 13.1 Example: Simple Harmonic Oscillator

As an example of a system of ODEs, consider a harmonic oscillator, which is characterized by an equilibrium position and a restoring force proportional to the displacement with friction. The system state will be a pair $$y = (y_1, y_2)$$ representing position and momentum: a point in phase space. The change in the system with respect to time is given by the following differential equations.25

$\frac{d}{dt} y_1 = y_2 \qquad \frac{d}{dt} y_2 = -y_1 - \theta y_2$

The state equations implicitly define the system state at a given time as a function of an initial state, elapsed time since the initial state, and the system parameters.

### Solutions Given Initial Conditions

Given a value of the system parameter $$\theta$$ and an initial state $$y(t_0)$$ at time $$t_0$$, it is possible to simulate the evolution of the solution numerically in order to calculate $$y(t)$$ for a specified sequence of times $$t_0 < t_1 < t_2 < \cdots$$.

1. This example is drawn from the documentation for the Boost Numeric Odeint library (Ahnert and Mulansky 2011), which Stan uses to implement the rk45 solver.