An Application: Decoupling a System of ODEs

System of ODEs

Let

and

be functions of

that satisfy the following system of ordinary differential equations (ODEs):

where

means

for

. The three variable functions are said to be coupled to each other. First of all, we rewrite the above system as a matrix equation:

, or more compactly,

. We diagonalize

such that

We make a change of variables:

. It is easy to see that

. Therefore,

This system of ODEs is very easy to solve as it is decoupled:

, where

are constants. Hence, we change the variables back to

and we have

Solve the following system of ODEs: