What’s the connection between differential equations and eigenfunctions/eigenvalues?

I’ve read differential equations can be understood as “eigenproblems” (using linear algebra concepts, basically), but I’m struggling to see just how.

Could somebody help me out in this? I’m aware it’s a very general question, but even a book suggestion or online reference would do wonders for me.

Thanks in advance!




Have you taken a course of Differential equations? Because that answers your question.
– imranfat
Oct 20 at 21:18



E.g. you might look at these notes of mine
– Robert Israel
Oct 20 at 21:31



x′=Ax,x(0)=x0x’=Ax,x(0)=x_0 is easy to solve if x0x_0 is an eigenvector of AA because each component satisfies the 1D ODE x′=λxx’=\lambda x. We build the general solution to x′=Axx’=Ax this way, taking advantage of the superposition principle. In the process we can also solve nth order linear ODEs with constant coefficients (since these are just a special case of x′=Axx’=Ax, after a suitable change of variable). Linear ODEs with nonconstant coefficients can have the same property (but finding the eigenfunctions is more difficult).
– Ian
Oct 20 at 21:48