When you are changing the indexing, set the exponents of x equal to k.
; ; ; and
Now all you exponents are just k.
When you solve for n, you simple sub what n equals in for n of the giving sum.
For instance, in the first sum,
I used these:
I Substitued the values in the origional equation:
Next, I multiplied the in, and the :
Now, I need all of these with the same starting value for and have to the same power.
*Here is where i get messed up*
The last term in is sum has the not , so did I do this wrong or how do I find the recursive relation?
Questions:
How do i find the recursive relation?
What is the recursive relation?
This is my FIRST time trying to use power series with differential equations so in the explanation if you could explain everything with a lot of detail(not skipping steps) that would be great.
EDIT: I used http://www.stewartcalculus.com/data/...lveDEs_Stu.pdf for reference.
Mr. Fantastic, who is an admin, deletes my post if I post the solution.
I made a mistake in my first comment but here is what you need to do.
In your final sum where you have all and one , say and .
Sub into summations.
You will now have 3 and .
However, by making the sub, you will now have the final and all the other xs will be of the k power.
For the 3 sums that don't start at 1, just take the first term of each one. this bump them from to .
Here is an example so you see:
That was your first sum. Now do that for all others that start at 0.
THANKS!!! Now I have one last question...I hope.
How do I: Use the recursion relation to find the series expansion through for and
Here are my recursion relations that I found by plugging the n values from 0 to 3 in my recursion relation. Then I got them all in terms of just and . From here how do i find the series expanion mentioned above?:
For
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For
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For
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For