I Y'' + y = 0 solution and recursion relation

3
0
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).

I've also found a recursion relation for the equation to be:

An+2 = -An / (n+2)(n+1)

I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?

Any help would be greatly appreciated!
 

hilbert2

Science Advisor
Insights Author
Gold Member
1,246
379
So is this the recurrence relation for the coefficients ##c_i## in the power series representing the solution: ##y(x)=c_0 + c_1 x + c_2 x^2 \dots## ? It shouldn't be difficult to take the sum of the power series of sine and cosine multiplied by a constant, and then show that the result satisfies both the DE and that recurrence relation.
 

vela

Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,411
1,099
Try calculating the first few terms of the series in terms of ##a_0## and ##a_1##. Hopefully, you'll recognize the resulting two series.
 
<Moderator's note: Approved as it is more than two weeks since the OP has been seen. Member has been warned not to post full solutions. This is an exception as it closes the thread.>

Let's start from your reccurence relation:

##A_{n+2}=\frac{-A_N}{(n+2)(n+1)}##

First collect even ##n## values:

##A_2= \frac{-A_0}{2.1}##

##A_4= \frac{-A_2}{4.3}=\frac{A_0}{4.3.2.1}=\frac{A_0}{4!}##

##A_6= \frac{-A_4}{6.5}=\frac{-A_0}{6.5.4.3.2.1}=\frac{-A_0}{6!}##

.........

Now take odd n values:

##A_3= \frac{-A_1}{3.2}##;

##A_5= \frac{-A_3}{5.4}=\frac{A_1}{5.4.3.2.1}=\frac{A_0}{5!}##

##A_7= \frac{-A_5}{7.6}=\frac{-A_0}{7.6.5.4.3.2.1}=\frac{-A_0}{7!}##

.........

So final solution is


On putting the values of ##A_n## in Maclaurin series solution (##y(x)=\sum_{n=0}{ A_n x^n}##),


##y(x)=\sum_{n=0}{ \frac{(-1)^n x^{2n}}{ 2n!} + \frac{(-1)^n x^{2n+1}}{ 2n+1!}}##

##y(x)= A_0 \cos (x) + A_1 \sin (x)##
 
Last edited by a moderator:

Want to reply to this thread?

"Y'' + y = 0 solution and recursion relation" You must log in or register to reply here.

Related Threads for: Y'' + y = 0 solution and recursion relation

Replies
3
Views
3K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
4
Views
1K
  • Posted
Replies
1
Views
2K
Replies
3
Views
3K
Replies
5
Views
731

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top