Second Order Linear ODE - Power Series Solution to IVP

  • #1
ChemistryNat
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Homework Statement


Let y(x)=[itex]\sum[/itex]ckxk (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)=[itex]\sum[/itex]ckxk (k=0 to ∞)
y'(x)=[itex]\sum[/itex](kckxk-1) (k=1 to ∞)
y''(x)=[itex]\sum[/itex](k(k-1))ckxk-2 (k=2 to ∞)

The Attempt at a Solution



(x2-1)[itex]\sum[/itex](k(k-1))ckxk-2 (k=2 to ∞) +[itex]\sum[/itex](kckxk) (k=1 to ∞)+[itex]\sum[/itex]ckxk (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[itex]\sum[/itex]ckxk (k=0 to ∞) where C0=0, or 2[itex]\sum[/itex]ckxk (k=1 to ∞)
but I'm not sure I can even do that mathematically?Thank you!
 
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  • #2
ChemistryNat said:

Homework Statement


Let y(x)=[itex]\sum[/itex]ckxk (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)=[itex]\sum[/itex]ckxk (k=0 to ∞)
y'(x)=[itex]\sum[/itex](kckxk-1) (k=1 to ∞)
y''(x)=[itex]\sum[/itex](k(k-1))ckxk-2 (k=2 to ∞)

The Attempt at a Solution



(x2-1)[itex]\sum[/itex](k(k-1))ckxk-2 (k=2 to ∞) +[itex]\sum[/itex](kckxk) (k=1 to ∞)+[itex]\sum[/itex]ckxk (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[itex]\sum[/itex]ckxk (k=0 to ∞) where C0=0, or 2[itex]\sum[/itex]ckxk (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?
 
  • #3
Dick said:
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?
 
  • #4
ChemistryNat said:
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.
 
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  • #5
Dick said:
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?
 
  • #6
ChemistryNat said:
Oh! so I can just leave it be and put it with the other x1 coefficient terms?

Sure.
 
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