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 : So 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