A coupled system is formed of two differential equations with two dependent variables and an independent variable.

An example -

where a, b, c and d are given constants, and both y and x are functions of t.

How do we solve coupled linear ordinary differential equations?

Use elimination to convert the system to a single second order differential equation. Another initial condition is worked out, since we need 2 initial conditions to solve a second order problem. Solve this equation and find the solution for one of the dependent variables (i.e. y or x). Use this solution to work out the other dependent variable.

For example:

How do we solve

(1)

(2)

with initial conditions and ?

Step 1: First make x the subject of (1), .

Step 2: Substitute in (2) to get which simplifies to with initial conditions and .

Step 3: The roots of the auxiliary equation are 2, 1. Hence the solution to the homogeneous problem is .

Step 4: Substituting the initial conditions gives i.e. .