Methods for transforming from coupled differential equations to a single
n^{th} order differential equation were discussed on the page "System
Representation by Differential Equations,"
example 3 and
example 4. Another
example is included below. It shows how to start with a set of
coupled differential equations and transform them into a single n^{th}
order differential equation.

Consider the system shown with f_{a}(t) as input and x(t) as
output. Find the differential equation relating x(t) to f_{a}(t).

We can write free body equations for the system at x and at y.

Freebody Diagram |
Equation |

Take the Laplace Transform of both equations with zero initial conditions (so derivatives in time are replaced by multiplications by "s" in the Laplace domain).

Now solve the second equation for Y(s) and substitute into the first equation and clear the fractions (so there are only positive powers of s).

Convert back to differential equation (replacing "s" in Laplace by a derivative in time).

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College