Power Series: Question about constants

[Saladsamurai]

Homework Statement

So I have attached the problem in image:

Pr and A are just numbers (constants) that are given. I solved the equation by power series solution. However, I am just confused because it is a second order DE but I only have one arbitrary constant of integration a_o. I am not sure where I went wrong? Should I have defined BOTH a₀ and a₁ to be arbitrary? I never got the hang of the little details with series solutions; perhaps I can get some input on this.

Thanks

Also, I would be interested to hear any other ideas about ways to solve this equation, but my power series question is my main concern.

 

[Mark44]

I didn't look very closely, but it looks like you're on the right track. Since your equation is second order, you will have two arbitrary constants, which in your case are a₀ and a₃. Problems like this have two initial conditions y(x₀) = y₀, and y'(x₀) = y₁. Since your series is a Maclaurin series, x₀ = 0.

 

[LCKurtz]

Look at the step where you change the index so it starts at n = -2. The other two sums start at n = 0 so they can't be grouped in the sum at that step. Furthermore, n = -1 and n = -2 don't give anything in the first sum and n= 0 doesn't give anything in the second. So write that step:

$$\sum_{n=0}^\infty(n+1)(n+2)a_{n+2}x^n+A\sum_{n=1}^{\infty}na_nx^n -A\sum_{n=0}^\infty a_nx^n$$

Now n = 0 will give a₂ in terms of a₀ and n = 1 will give a₃ in terms of a₁, with a₀ and a₁ arbitrary.

 

[Saladsamurai]

Sweet! I went through it again and got a0 and a1 as my arbitrary constants. Out of curiosity, anyone see another way to solve this? I was thinking that if we divide through by y' we would have:

y"/y' + Ax - Ay/y' = 0

Now the first term can be written ln[y']'. If I could write the last term in a similar fashion (i.e. as a derivative), I could integrate directly.