Solving DE using power series (w/initial conditions)

[Fuzedmind]

Homework Statement

Solve the initial value problem y'' = y' + y where y(0) = 0 and y(1) = 1

derive the power series solution y(x) = $$\ \ \sum_{n=1}^{\infty}{(F_{n}x^n)/n!} \ \$$ where {F_n} is the sequence 0,1,1,2,3,5,8,13... of Fibonacci numbers defined by F₀ = 0 and F₁ = 1

Homework Equations

The Attempt at a Solution

I plugged in the series and got this equation:

$$\ \ \sum_{n=2}^{\infty}{(n)(n-1)c_{n}x^(n-2)} \ \ + \ \ \sum_{n=1}^{\infty}{c_{n}x^(n-1)} \ \ + \ \ \sum_{n=0}^{\infty}{c_{n}x^n} \ \ = x$$

from which i got the recurrence equation:

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

Problem is I don't know how to get F₀ and F₁ from the initial conditions. My teacher never taught us how to solve initial condition power series ODE's nor are there any examples in the book. Can anyone explain it to me?

 

[LCKurtz]

You are assuming y = c₀ + c₁x + c₂x²+...

y(0) is just c₀. And y'(0) would be c₁ , which would be the usual conditions for an IVP. Are you sure you have the second boundary condition written correctly? My guess it is supposed to be y'(0) = 1.

 

[Fuzedmind]

no y(0) = 0 and y(1) = 1, I am looking at it right now

 

[LCKurtz]

I don't have time to work through the solution right now, but what you have then is a boundary value problem, not an initial value problem as claimed. I'm suspecting a misprint. You might just try using y'(0)=1 and working it through to see if you get the correct answer.

 

[LCKurtz]

I had a little time later in the day. Your text indeed has a misprint. If you take your equation with the initial conditions y(0) = 0 and y'(0) = 1, which give you the values of c₀ and c₁ and work out the series solution, you get exactly what the problem claims you will get.

 

[Fuzedmind]

Ok thanks a bunch man. I did that and it worked out for me too.