Power Series Solution of Differential Equation

  • #1

Homework Statement


(x^2)y' = y


Homework Equations





The Attempt at a Solution


Plugging in series everywhere I get the equation [itex]\sum na_{n}x^{n+1}[/itex] = [itex]\sum a_{n}x^{n}[/itex]. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also do not know how to handle the sum on the left starting at the x^2 term while the one on the right starts at the constant term.

I hope I made the statement of the problem clearer.
Thanks in advance
 
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Answers and Replies

  • #2
Ray Vickson
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Homework Statement


(x^2)y' = y


Homework Equations





The Attempt at a Solution


I replace y and y' with power series and combine them into one series by messing with the indices, but the resulting recursion relation is not correct?

I cannot figure out whether you are making a statement or asking a question; the question mark at the end of your sentence is throwing me off. Anyway, if you show your work we will be better able to comment.
 
  • #3
I have edited the write up to hopefully make it more clear
 
  • #4
pasmith
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695

Homework Statement


(x^2)y' = y


Homework Equations





The Attempt at a Solution


Plugging in series everywhere I get the equation [itex]\sum na_{n}x^{n+1}[/itex] = [itex]\sum a_{n}x^{n}[/itex]. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also do not know how to handle the sum on the left starting at the x^2 term while the one on the right starts at the constant term.


I hope I made the statement of the problem clearer.
Thanks in advance

[tex]x^2 \frac{d}{dx} \sum_{n=0}^\infty a_nx^n = x^2 \sum_{n=0}^\infty na_nx^{n-1}
= \sum_{n=0}^\infty na_nx^{n+1} = \sum_{n=1}^\infty na_nx^{n+1}
[/tex]
where the last follows because [itex]n = 0[/itex] when [itex]n = 0[/itex], and substituting [itex]k = n +1[/itex] is the next step.

However solving the ODE directly gives
[tex]
y(x) = A\exp\left( -\frac 1{x}\right)
[/tex]
where we have the problem at zero that
[tex]
\lim_{x \to 0^{+}} y(x) = \lim_{x \to 0^{+}} y^{(n)}(x) = 0
[/tex]
for all [itex]n \geq 1[/itex], so there is no convergent power series about zero unless [itex]A = 0[/itex].

For [itex]x < 0[/itex] it's even worse:
[tex]
\lim_{x \to 0^{-}} y(x)
= \begin{cases} \infty & A > 0 \\ 0 & A = 0 \\ -\infty & A < 0
\end{cases}
[/tex]
 
  • #5
Sorry I'm still confused. I got the series you got for the y' term and then I changed it to [itex]\sum^{\infty}_{n=2}(n-1)a_{n-1}x^{n}[/itex] so I could combine it with the other sum, which I rewrote as a[itex]_{0}[/itex]+a[itex]_{1}[/itex]x+[itex]\sum^{\infty}_{n=2}a_{n}x^{n}[/itex]. Normally I would set all coefficients equal to zero and then find a recursion relation. This did not work this time. If I've understood you're post there is no convergent power series so this method won't work. What should I do differently?
 
  • #6
HallsofIvy
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What you have, initially, [itex]\sum_{n= 1}^\infty na_n x^{n+ 1}= \sum_{n= 0}^\infty a_nx^n[/itex].
(Notice that the left sum starts at n= 1. That is because if when n= 0, the term is "[itex]0x^0= 0[/itex].) You say you tried "to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer" but you don't say what you tried what answer you got or what the correct answer you should have gotten.

You could try writing out the first few terms of each sum: [itex]a_1x^2+ 2a_2x^3+ 3a_3x^4+ \cdot\cdot\cdot[/itex] on the left and [itex]a_0+ a_1x+ a_2x^2+ a_3x^3+ a_4x^4+ \cdot\cdot\cdot[/itex] and since those are equal we must have [itex]a_0= 0[/itex], [itex]a_1= 0[/itex], [itex]a_2= a_1[/itex], [itex]a_3= 2a_2[/itex], [itex]a_4= 3a_3[/itex], etc.

Generally, we can compare corresponding coefficients for [itex]\sum_{n= 1}^\infty na_n x^{n+ 1}= \sum_{n= 0}^\infty a_nx^n[/itex] as you do by letting j= n+ 1 in the left sum and j= n in the sum on the right:
[itex]\sum_{n= 1}^\infty na_n x^{n+ 1}= \sum_{j= 2}^\infty (j-1)a_{j-1}x^j= \sum_{j= 0}^\infty a_j x^j[/itex]. Since the left sum does not begin until j= 2, for j= 0 we have [itex]a_0= 0[/itex] and for i= 1,[itex]a_1= 0[/itex]. For j> 1, [itex](j-1)a_{j-1}= a_j[/itex]. In particular, [itex]a_2= 1a_1= 0[/itex], [itex]a_3= 2a_2= 2(0)= 0[/itex], etc.
 
  • #7
Ray Vickson
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Sorry I'm still confused. I got the series you got for the y' term and then I changed it to [itex]\sum^{\infty}_{n=2}(n-1)a_{n-1}x^{n}[/itex] so I could combine it with the other sum, which I rewrote as a[itex]_{0}[/itex]+a[itex]_{1}[/itex]x+[itex]\sum^{\infty}_{n=2}a_{n}x^{n}[/itex]. Normally I would set all coefficients equal to zero and then find a recursion relation. This did not work this time. If I've understood you're post there is no convergent power series so this method won't work. What should I do differently?

The solution y(x) is non-analytic at x = 0; that means that y(x) is not equal to its Taylor series about x = 0 (i.e., Maclauren series), even though that Maclauren series is convergent on the whole real line! The solution cannot be represented as a series of the form ##\sum_{n=0}^{\infty} c_n x^n##. Your power series method yields the correct Maclauren series for y(x), but that is not y(x) itself.
 
  • #8
Ok this made a lot of sense. Thank you both very much.
 
  • #9
Ray Vickson
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Ok this made a lot of sense. Thank you both very much.

Actually, my previous post is not quite accurate. There are many solutions, depending on conditions at a particular x0 ≠ 0. Your series solution finds one of them (y(x) = 0 for all x); the others cannot be expanded as power series around x = 0.
 
  • #10
HallsofIvy
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I assume that you are required to use series. A much simpler solution is to separate variables to get
[tex]\frac{dy}{y}= \frac{dx}{x^2}[/tex]
so integrating [itex]ln|y|= -\frac{1}{x}+ C[/itex].

And then [itex]y= C' e^{-1/x}[/itex].

Perhaps you can use that as a check.
 

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