Series solution to linear differential equations

[sunda]

Homework Statement

Given the ODE y''-ty'+y=0 where y(0)=1 and y'(0)=0
Assume y(t)=Ʃn=0 ( a(n) t^n ) (power series centered at 0)

find the general form of the solution ( an=f(n) )

The Attempt at a Solution

I used the initial conditions to determine the values a0=1 and a1=0

Determined the recurrence relation a(n+2)= a(n) (n-1) / (n+1)(n+2)

And found the first six non-zero terms (which was asked for in an earlier part of the question.

a(0)=1 a(1)=0 a(2)=-1/2! a(3)=0 a(4)=-1/4! a(5)=0 a(6)=-3/6! a(7)=0 a(8)=-15/8! a(9)=0 a(10)=-105/10!

I am having a really tough time coming up with a(n) I can identify a few patterns such as there is obviously a component 1/n! and since all the odd terms are 0 it would be 1/(n+1)! I attempted to handle the first term being positive and the rest negative using (n-1)/abs(n-1) I am assuming there is a better way to handle this however. I am very lost on coming up with a series representation of the numerator {1-1-1-3-15-105-...-...}

Any help will be appreciated very much.
Thanks for your time!

 

[mighty2000]

Thank you for your question. Your approach so far looks correct, and it seems like you are on the right track. Here are a few tips to help you determine the general form of the solution:

1. Look for a pattern in the coefficients: As you have already noticed, the coefficients of the series seem to alternate between positive and negative values. Additionally, the values of the coefficients also seem to be increasing, with each odd term being 0. This suggests that the general form of the solution may involve a combination of factorials and alternating signs.

2. Use a summation notation: Instead of trying to find a closed form expression for the coefficients, you can use a summation notation to represent the series. This will make it easier to manipulate and work with the series, and may also help you identify a pattern. For example, the series can be written as a summation from n=0 to infinity of (-1)^n (2n)!/((2n+1)! n!).

3. Consider using a generating function: A generating function is a powerful tool that can help you find a closed form expression for a series. In this case, you can use the generating function for the series to determine the general form of the solution. The generating function for the series is given by f(x) = exp(-x^2/2).

I hope these tips will help you in determining the general form of the solution. Keep in mind that solving differential equations can be a challenging task, so don't get discouraged if you are having trouble. Keep exploring different methods and approaches, and don't be afraid to ask for help. Best of luck with your work!