Coupled Systems

What is a coupled system?

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 \[ \frac{d^{2}y}{dt^{2}} - 3\frac{dy}{dt} + 2y = 0 \] with initial conditions \[ y(0) = 2 \] and \[ \frac{dy}{dt}(0) = - y(0) + 3x(0) = -2 + 3 = 1 \] .

Step 3: The roots of the auxiliary equation \[ m^2 - 3m + 2 = 0 \] are 2, 1. Hence the solution to the homogeneous problem is \[ y = Ae^{-t} + Be^{-2t} \] .

Step 4: Substituting the initial conditions gives \[ A = 5, B = -3 \] i.e. \[ y = 5e^{-t} - 3e^{-2t} \] .

Step 5: Now we have \[ x = \frac{1}{3}(\frac{dy}{dt} + y) = \frac{1}{3}(\frac{d}{dt}(5e^{-t} - 3e^{-2t}) + 5e^{-t} - 3e^{-2t}) = e^{-2t} \] . Hence the solution is \[ y = 5e^{-t} - 3e^{-2t} \] and .

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