Coragyps

That was my situation that whole damn semester of DiffE.

lpetrich

ex-xian
But they can also be written in terms of trig functions, right? Do you just choose whichever one is simplest?

Yes if one finds that convenient.
Yes.

Also, is there a similiar "method" for solving, I'm not sure if this is the correct terminology, second order DE?

Yes.

I know that there are different solutions for each, but is there any way to solve them when they are more complicated.

More complicated in what way?

And what about partial DE?

In fact, this approach works for any set of differential equations that satisfy these conditions:

* Linearity in the dependent variables and their derivatives (does not matter how many there are)

* Independent variables only appear as differentiators of the dependent variables (does not matter how many there are)

Tenpudo

This thread reminds me of just how much one can forget in 5 years without a math class...

wade-w

I never did like diffeq much. But I recall that I was taking linear algebra the same term. At least once a week while in my de class, when going over some theorem, I would think to myself, "Wait a minute, that's just a special case of the theorem we did this morning!"

IIRC, the prefered approach to solving diff eqs is not so much guessing, but recognizing the form, and applying the proper technique. Much like integration.

I suspect it should be easier to do ordinary diffeq using linear algebra techniques. But I could be wrong, since I never bothered to research the idea. Set theory is so much more interesting!

Undercurrent

True. An like integration, there is a technique based on Lie point symmetries that will tell you if a given ODE has a closed-form solution, and the change of variables required to get it there, but it is painful to use, so one usually just sticks to the guessing.

There are techniques based on Lin. Alg. for linear ODEs (e.g. Green's functions are essentially continuous matricies.) I have yet to see much in non-linear ODEs along those lines, though.

Shake

Just wait 'til it's been another 6 years beyond that!

wade-w

My idea was primarily based on the observation that the differentiable functions are a vector space. Which, I'm sure, is why the solution for linear differential equations lpetrich refers to works.

And its been over 20 years since I took a math course, Shake. IIRC, I had diffeq in 1978.

xorbie

I'm just entering college this fall, and I'm gonna be getting a double major in physics and math, so this thread seems right up my ally. I guess I really don't have a point, other than to say that this healps me appreciate how tough it is gonna be for me to learn all of this (plus some more) in 4 years.