10 | Ordinary Differential Equations (ODEs)

In this section, we use MATLAB to numerically solve ordinary differential equations (ODEs) and systems of ODEs. We will take advantage of MATLAB’s built-in solvers such as ode45.

At the end of this section you should be able to:

Describe what an Ordinary Differential Equation is
Place an ODE or system of ODEs into State-Space form
Utilize ode45 to numerically solve a state-space model
Use event detection to stop an ODE solution or modify it when events trigger

Solving ODEs By Hand

Numerical Solutions of ODEs

State Space Form

Applying ODE45

Physics Examples

Using Event Detection: continuing the cannonball example but stopping the ODE solver once the ball hits the ground using event detection in ODE45.

Lecture Code

Additional Resources

Solving ODEs By Hand

Analytical Solution to an ODE I: here we go through the steps of analytically solving a homogeneous first-order linear separable differential equation.

Analytical Solution to an ODE II: here we go through the steps to solve a homogeneous second-order linear differential equation

Numerical Solutions of ODEs

State Space Form

Solving ODEs numerically: when we cannot get an analytic solution to a differential equation or want to automate the process, we can instead solve for the dependent variable(s) numerically. In Matlab and most ODE solvers, we first need to put our differential equation(s) into state space form.

State Space Example 1: we take a second-order linear nonhomogeneous differential equation and rewrite it in state space form as two first-order differential equations.

State Space Example 2: we take two first-order non-linear nonhomogeneous differential equation and rewrite it in state space form as two first-order differential equations.