# Solving a Linear ODE using a power series

Gold Member

Power series
ODE

## The Attempt at a Solution

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Sorry for not typing all those things out from my phone..
How can I get C1?
And how can I put the solution in the required format? (I dont know how to put it in summation sign... and i cannot even solve out for r in the question....)
Thank you so much for any help!

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## Answers and Replies

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BvU
Homework Helper
2019 Award
cannot even solve out for r in the question
Small wonder: you leave out ##r## from the derivatives of ##y##. Except in ##C_n##, you want to replace ##n## by ##n+r##

Gold Member
Thank you.
However even i used r+n, i still cannot make it into the required form... did i have something wrong with it? Or what should I do next to obtain the answer in the required format? Thank you.

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ehild
Homework Helper
You have a mistake in the equation for r. It should be r2+4r+4=0

BvU
Gold Member
Thanks for pointing this out...

r=-2 which is now correct,
However, i cannot solve out the denominator of the summation part... how can I write it in form of summation?
Thank you very much!

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ehild
Homework Helper
Substitute r = -2 and simplify the denominator of an.

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I have already substituted r=-2, but i dont know the recurrance relation of that denominator... do you mind to give me some hints or relationship related?
Thank you

ehild
Homework Helper
Type in the recurrence relation of an, please. Your handwriting is blurred,
I have already substituted r=-2, but i dont know the recurrance relation of that denominator... do you mind to give me some hints or relationship related?
Thank you
You got the recurrence relation for an:
$$a_n=\frac{4 a_{n-1}}{(n+r)(n+r-1)+5(r+n)+4}$$
Substitute r=-2 and simplify. What do you get?

Gold Member
Sorry for blur image... let me post a clearer image here:

I have got my answer as 4^n/(n!), yet it is wrong as by the submission system... whats wrong with my answer? Thank you

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ehild
Homework Helper
Sorry for blur image...
View attachment 223115
I have got my answer as 4^n/(n!), yet it is wrong as by the submission system... whats wrong with my answer? Thank you
The recurrence relation for an is

##a_n=\frac {4 a_{n-1}}{n^2}##, with a0=1.
How did you get n! in the numerator?

Gold Member
As I have to write the answer in terms of n only as a summation form (refer to the question image I have posted in #1), only writting it in recurrance relationship is not enough...
As the image I posted in #9 post explained how i got “n!”, when n tends larger, the relationship a(n) tends to be n! in denominator

4^n*(n!)/(n!)^2 =4^n/(n!)

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Gold Member
I have already tried my best in obtaining the answer, do you mind to lead me what should I write in the blank as of the question image posted in post #10 please?
Thank you so much for your help!

ehild
Homework Helper
I have already tried my best in obtaining the answer, do you mind to lead me what should I write in the blank as of the question image posted in post #10 please?
Thank you so much for your help!
I do not understand what you mean on "summation form"
You need to find the coefficients cn in the power series solution.
You derived the recurrence relation $$c_n=\frac{4}{n^2}c_{n-1}$$, but you wrote wrong results for c1, c2, c3, c4...
c0=1
$$c_1=\frac{4}{1^2}c_{0}= \frac{4}{1^2}$$
$$c_2=\frac{4}{2^2}c_{1}=\frac{4^2}{2^2\cdot1^2}=\frac{4^2}{(2!)^2}$$
$$c_3=\frac{4}{3^2}c_{2}=??$$
and so on. So what is cn?

yecko