How can a differential equation be expressed as a series?

Question

I missed this lecture, and I'm really ticked off about it. Apparently, diff. EQ's can be expressed in terms of a series. Using a very simple diff. EQ, such as y' - y = 0, please explain how you would rationalize this idea.

Answer 1

Depending on your level of diff eq...

any equation can be listed as a series expansion...and it may be sufficient (or close enough) to take the integral of "enough" terms and sum up the integrals...(the integral of the sums is the sum of the integrals).

this is a way of approximating when the higher order terms don't play a big role.

This is done in the engineering world quite a bit where the first couple of terms dominate the system (of which the diff eq is describing). Think of a tall building (or bridge) and the wind forces upon it. Steady winds will push on the building and make it sway but if the wind changed very rapidly, the building will sort of "dampen" out the higher order (higher frequency) inputs.

Another classic example (and one that every 4 year old has most likely mastered) is the swing. If the person pushing the swing (or the child 'pumping his legs) does it very quickly the response of the swing won't be high, it is much more advantageous to push (or pump) at a much slower frequency (corresponding to lower orders).

Hopefully, this is where you are coming from...I didn't know if you wanted a rational explanation for the idea or to delve into the mathematics.

Answer 2

First, *solutions* of differential equations can be written in terms of power series. This is done as follows for y'=y:

Assume
y=a_0 + a_1 x + a_2 x^2 + a_3 x^3 +...

Then, taking a derviative,
y'=a_1 +2a_2 x +3a_3 x^2 +4 a_4 x^4+...

Now set up y'=y and equate coefficients:
a_0=a_1
a_1=2a_2
a_2=3a_3
a_3=4a_4
etc.

Now solve in terms of a_0:
a_1=a_0, a_2=(1/2)a_0, a_3 =(1/6)a_0, ...

Now write the solution:
y=a_0 (1+x+(1/2)x^2 +(1/6) x^3 + (1/24) x^4 +....).

This is the power series solution. You might be able to recognize it as y=a_0 exp(x).

For differential equations of degree two or more, you won't be able to solve for all of the coefficients in terms of a_0. You will need more than one arbitrary constant for the general solution.

Answer 3

well This was one of the hardest sections In this course. Trust me it's not easy explaning it like this, you're better of going to your prof and asking for help

Answer 4

You can use Maclaurin's or Taylor's series.

Answer 5

see any differential equation book

Answer 6

http://tutorial.math.lamar.edu/AllBrowsers/3401/Series.asp

Good luck to you.