Solving a Linear ODE using a power series

  • Thread starter yecko
  • Start date
  • #1
yecko
Gold Member
260
6

Homework Statement


45F8F685-B056-4297-9F96-5679CA7A39EC.jpeg


Homework Equations


Power series
ODE

The Attempt at a Solution


1B4BB3F1-8717-4BB3-9B73-9668905861F9.jpeg
[/B]

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!
 

Attachments

Last edited:

Answers and Replies

  • #2
BvU
Science Advisor
Homework Helper
2019 Award
13,429
3,198
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##
 
  • #3
yecko
Gold Member
260
6
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.
7AD2046A-E29F-40D7-A423-DE4EDFF2A2D2.jpeg
 

Attachments

  • #4
ehild
Homework Helper
15,510
1,897
You have a mistake in the equation for r. It should be r2+4r+4=0
 
  • Like
Likes BvU
  • #5
yecko
Gold Member
260
6
Thanks for pointing this out...
5A68D7FB-3FFF-40CB-859B-1E6F9B9DCE44.jpeg

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!
 

Attachments

  • #6
ehild
Homework Helper
15,510
1,897
Substitute r = -2 and simplify the denominator of an.
 
  • #7
yecko
Gold Member
260
6
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
 
  • #8
ehild
Homework Helper
15,510
1,897
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?
 
  • #9
yecko
Gold Member
260
6
Sorry for blur image... let me post a clearer image here:
B63C09BC-82B5-41CB-8B36-0376F400634F.jpeg

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
 

Attachments

Last edited:
  • #11
ehild
Homework Helper
15,510
1,897
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?
 
  • #12
yecko
Gold Member
260
6
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!)
 
Last edited:
  • #13
yecko
Gold Member
260
6
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!
 
  • #14
ehild
Homework Helper
15,510
1,897
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?
 
  • Like
Likes yecko

Related Threads on Solving a Linear ODE using a power series

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
0
Views
2K
Replies
2
Views
3K
Replies
4
Views
855
  • Last Post
Replies
16
Views
1K
Replies
4
Views
2K
Replies
3
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
754
Replies
5
Views
2K
Top