Differential Equations Workbook For Dummies®

11.5. Solving Differential Equation Systems

When you have matrices and determinants, as well as eigenvalues and eigenvectors, mastered (work through the problems in the previous sections if you still need practice), you're ready to solve systems of linear first order differential equations.

Take a look at this system of homogeneous differential equations:

y₁′ = y₁ + y₂

y₂′ = 4y₁ + y₂

These differential equations are linked, which means they both contain y₁ and y₂, and therefore must be solved together. You can write them in this form:

which you can then write like this:

y′ = Ay

In this case, y′, A, and y are all matrices:

If A is a matrix of constant coefficients, then you can assume a solution of the form

y = ξe^rt

No, ξ isn't just a random symbol I've thrown in to see whether you're still awake. It actually stands for an eigenvector. Substituting your supposed form of the solution into the system of differential equations gives you

rξe^rt = Aξe^rt

Now you can subtract Aξe^rt from both sides to get

(A − rI)ξe^rt = 0

or

(A − rI)ξ = 0

Tada! You've just found the equation that specifies ...