TITLE: How are eigenvectors/eigenvalues and differential equations connected?
QUESTION [18 upvotes]: In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that both are connected in some way, i.e. there is an eigenvector/eigenvalue approach  to solve differential equations. Unfortunately most of the text I find are way beyond my understanding and seem to require very extensive study.
Is there a possibility to provide me with an intuition of the connection anyway and perhaps give an easy example of how to use this approach to solve a differential equation?

REPLY [2 votes]: This is the best explanation I have seen so far. This short paper not only explains the connection between eigenvalues, eigenvectors and differential equations using very clear, undergraduate math but also has lots of visualizations... and nice examples concerning the development of Romeo's and Juliet's love:
http://wcherry.math.unt.edu/math2700/diffeq.pdf