Relationship between Linear Algebra and Differential equations

[discoverer02]

I just came from a class lecture that tied together the relationship between linear algebra and differential equations. The lecture dealt only with homogeneous linear equations. I understood about 90% of it and want to try to tie together the loose ends.

In a nutshell, if I have a homogeneous linear differential equation of degree n, where L is a linear differential operator of order n. Then the general solution of the homogeneous linear differential equation is the linear combination of n linearly independent elements of ker(L).

I haven't seen this applied to an example yet, so it's not entirely clear, but have I stated the relationship correctly?

I guess I'll see examples tomorrow, but I'd like to go into class with a crystal clear picture, so I can following along with what will probably be another lightning quick lecture.

Can anyone provide a simple example?

Thanks.

 

[discoverer02]

Nevermind.

We went over everything once more and finally went over some examples. Solving higher order homogeneous linear O.D.E.'s couldn't be easier. It's like a breath of fresh air after a month of sometimes grueling linear algebra.

 

[HallsofIvy]

Solving homogeneous linear O.D.E. with constant coefficients is as easy as solving a polynomial equation. I, personally, could imagine easier things than solving a fifth order homogeneous linear O.D.E with constant coefficients. You will probably soon have to deal with equations with variable coefficients!

 

[discoverer02]

You're absolutely right. I guess I was caught up in the moment. I really had to work hard to stay on top of the linear algebra that leads up to this revelation.

In the long run, I'm sure it only gets tougher.