Differential Equations - Power Series problem with e^t

[skyturnred]

Homework Statement

The problem is to solve:

y''+ty'+e^{t}y=0, y(0)=0 and y'(0)=-1

Homework Equations

The Attempt at a Solution

My main issue is the following: I normally find the recursion relation, and then factor out the t^{whatever} and I know that the coefficient to this must equal 0. However, the addition of the e^{t} means I can't factor out ALL of the t^{whatever} so I can't exactly equate the coefficient to 0.

So specifically with the question above, I get the following:

ty'=\sum^{inf}_{k=1}ka_{k}t^{k}

e^{t}=\sum^{inf}_{k=0}t^{k}/k!

e^{t}y=\sum^{inf}_{k=0}t^{2k}a_{k}/k!

y''=\sum^{inf}_{k=2}k(k-1)a_{k}t^{k-2}

Then I plug those into the DE, write out the first few terms so that they all start at k=2, and put them all under the same sum to get as follows:

a_{0}+a_{1}t+a_{1}t^{2}+2a_{2}+6a_{3}t+\sum^{inf}_{k=2}[(k+2)(k+1)a_{k+2}+ka_{k}+a_{k}t^{k}]t^{k}=0

From this you find that a_{2}=-a_{0}/2

and

a_{1}=0

and

a_{3}=0


Like I said, normally at this point I find the recursion relation and factor out a t^{k} so that I know that the entire coefficient is 0. But in this case, after factoring out the t^{k} one of the terms in the coefficient still have a t^{k}, so I don't know if I can't just equate this whole thing to 0.

Can anyone help?

Thank-you

 

[LCKurtz]

Homework Statement

The problem is to solve:

y''+ty'+e^{t}y=0, y(0)=0 and y'(0)=-1

Homework Equations

The Attempt at a Solution

My main issue is the following: I normally find the recursion relation, and then factor out the t^{whatever} and I know that the coefficient to this must equal 0. However, the addition of the e^{t} means I can't factor out ALL of the t^{whatever} so I can't exactly equate the coefficient to 0.

So specifically with the question above, I get the following:

ty'=\sum^{\infty}_{k=1}ka_{k}t^{k}

You don't need to put the itex tags around every item. Just put them around the whole equation. Right click on the above expression and see how I changed it for you.

e^{t}=\sum^{inf}_{k=0}t^{k}/k!

e^{t}y=\sum^{inf}_{k=0}t^{2k}a_{k}/k!

You don't multiply two series like that. You need the Cauchy Product. See:
http://planetmath.org/cauchyproduct