• Support PF! Buy your school textbooks, materials and every day products Here!

Power Series Solution to a Diff EQ

  • #1
2,981
2
[SOLVED] !!!Power Series Solution to a Diff EQ!!!

1. Homework Statement
Find the first 5 term of a Power series solution of

[tex]y'+2xy=0[/tex] (1)

Missed this class, so please bear with my attempt here.


3. The Attempt at a Solution

Assuming that y takes the form

[tex]y=\sum_{n=0}^{\infty}c_nx^n[/tex]

Then (1) can be written:

[tex]\sum_{n=1}^{\infty}nc_nx^{n-1}+2x\sum_{n=0}^{\infty}c_nx^n=0[/tex]

Re-written 'in phase' and with the same indices (in terms of k):

[tex]c_1+\sum_{k=1}^{\infty}(k+1)c_{k+1}x^k+\sum_{k=1}^{\infty}2c_{k-1}x^k=0[/tex]

[tex]\Rightarrow c_1+\sum_{k=1}^{\infty}[(k+1)c_{k+1}+2c_{k-1}]x^k=0[/tex]

Now invoking the identity property, I can say that all coefficients of powers of x are equal to zero (including [itex]c_1*x^0[/itex])

So I can write:

[itex]c_1=0[/itex] and

[tex]c_{k+1}=-\frac{2c_{k-1}}{k+1}[/tex]


Now I am stuck (I know I am almost there though!)

Should I just start plugging in numbers for k=1,2,3,4,5 ? Will this generate enough 'recursiveness' to solve for the 1st five terms?

Is that the correct approach?

Thanks!!
 

Answers and Replies

  • #2
2,981
2
Don't worry guys, I got it. And for those who might make future use of this thread, my approach was correct. Solving

[tex]
c_{k+1}=-\frac{2c_{k-1}}{k+1}[/tex]

For k=1,2....,9 generates enough coefficients to write out the first five nonzero terms of the solution by plugging them back into

[tex]
y=\sum_{n=0}^{\infty}c_nx^n[/tex]
 
Last edited:
  • #3
743
1
I think you need one more initial condition on your series (example [itex] y(0)=x_0[/itex]). You need to define [itex] c_0[/itex] (or you can just leave [itex] c_0 [/itex] as the 'integration' constant of the ODE). However, the recursion formula becomes pretty obvious once you have that. If you can't see it right away, try plugging in a few values. Since [itex] c_1=0[/itex] what can we say about all the odd labelled coefficients? You should get something that looks like the series of an exponential function. In fact, the series will have a closed form solution if you can see how your answer relates to the exponential series.
 

Related Threads for: Power Series Solution to a Diff EQ

Replies
4
Views
2K
Replies
1
Views
457
  • Last Post
Replies
2
Views
2K
Replies
1
Views
3K
  • Last Post
Replies
9
Views
809
Replies
3
Views
413
Replies
1
Views
2K
Top