Second Order Linear ODE - Power Series Solution to IVP

[ChemistryNat]

Homework Statement

Let y(x)=$\sum$c_kx^k (k=0 to ∞) be a power series solution of

(x²-1)y''+x³y'+y=2x, y(0)=1, y'(0)=0

Note that x=0 is an ordinary point.

Homework Equations

y(x)=$\sum$c_kx^k (k=0 to ∞)
y'(x)=$\sum$(kc_kx^k-1) (k=1 to ∞)
y''(x)=$\sum$(k(k-1))c_kx^k-2 (k=2 to ∞)

The Attempt at a Solution

(x²-1)$\sum$(k(k-1))c_kx^k-2 (k=2 to ∞) +$\sum$(kc_kx^k) (k=1 to ∞)+$\sum$c_kx^k (k=0 to ∞) -2x=0 ??

I'm not having an issue with the power series themselves, I'm just not sure how to incorporate in the "2x" term when I'm setting up the series equation. We didn't cover this scenario in class and I couldn't find anything like it in my textbook.

I've been trying to incorporate it as a series itself
ie. 2$\sum$c_kx^k (k=0 to ∞) where C0=0, or 2$\sum$c_kx^k (k=1 to ∞)
but I'm not sure I can even do that mathematically?Thank you!

 

[Dick]

Homework Statement

Let y(x)=$\sum$c_kx^k (k=0 to ∞) be a power series solution of

(x²-1)y''+x³y'+y=2x, y(0)=1, y'(0)=0

Note that x=0 is an ordinary point.

Homework Equations

y(x)=$\sum$c_kx^k (k=0 to ∞)
y'(x)=$\sum$(kc_kx^k-1) (k=1 to ∞)
y''(x)=$\sum$(k(k-1))c_kx^k-2 (k=2 to ∞)

The Attempt at a Solution

(x²-1)$\sum$(k(k-1))c_kx^k-2 (k=2 to ∞) +$\sum$(kc_kx^k) (k=1 to ∞)+$\sum$c_kx^k (k=0 to ∞) -2x=0 ??

I'm not having an issue with the power series themselves, I'm just not sure how to incorporate in the "2x" term when I'm setting up the series equation. We didn't cover this scenario in class and I couldn't find anything like it in my textbook.

I've been trying to incorporate it as a series itself
ie. 2$\sum$c_kx^k (k=0 to ∞) where C0=0, or 2$\sum$c_kx^k (k=1 to ∞)
but I'm not sure I can even do that mathematically?Thank you!

Your are going to equate powers of x to get a relation between the c_k values, right? The -2x will only contribute the x^1 term, yes?

 

[ChemistryNat]

Your are going to equate powers of x to get a relation between the c_k values, right? The -2x will only contribute the x^1 term.

So I can leave it in as a constant coefficient term, say 2C₁x? and then use that in combination with the other constants I've pulled out?

 

[Dick]

So I can leave it in as a constant coefficient term, say 2C₁x? and then use that in combination with the other constants I've pulled out?

Mmm. Sort of, but the coefficient your x^1 term is only going to have a -2 in it. Without any C_k term in front of it. It's just a constant.

 

[ChemistryNat]

Mmm. Sort of, but the coefficient your x^1 term is only going to have a -2 in it. Without any C_k term in front of it. It's just a constant.

Oh! so I can just leave it be and put it with the other x¹ coefficient terms?

 

[Dick]

Oh! so I can just leave it be and put it with the other x¹ coefficient terms?

Sure.