2nd order DE, is there a way to solve this without series?

  • Thread starter aeroegnr
  • Start date
  • #1
17
0
It looks simple enough:

y'' + x*y = x^2

However, I tried and I could not find a nontrivial solution to the homogeneous equation:

y'' + x*y = 0

Am I right in thinking you need to solve this with series?

No need to actually do it, I just need to know if it is possible otherwise (like variation of parameters or something else).
 

Answers and Replies

  • #2
37
0
can you use integrating factor?
 
  • #3
17
0
For a 2nd order equation? I know how to do that for first order but not second order equations.
 
  • #4
Dr Transport
Science Advisor
Gold Member
2,412
543
The homogeneous equation looks like the Airy equation.....
 
  • #5
17
0
Ah, so only power series is it then?

That's fine. It's just for some reason I thought I wasn't understanding some kind of trick to give a general solution.

It's almost like I saw an integral of
[tex]
\int e^x^2 dx
[/tex]

Which of course can only be done with power series, and thought I could integrate it and give a nice general solution.
 
  • #6
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
You could say that you get a "nice" general solution by dubbing it as Ai(x)..:biggrin:
 
  • #7
12
0
If y1 is known, you can use reduction of order to solve this.
 
  • #8
Dr Transport
Science Advisor
Gold Member
2,412
543
if memory serves me correctly, the Airy equation is proportional to a Bessel function of 1/3 order......Look out there online.
 
  • #10
selfAdjoint
Staff Emeritus
Gold Member
Dearly Missed
6,786
7
Has anyone tried this one in MAPLE? There might be a Bessel function integrating factor of the homogeneous equation (just interested, that's all, Max's link gives the answer).
 
  • #11
Mathematica gives:

Edit: Something too long or not properly formatted for PF to handle... But it was pretty much the Airy function.

cookiemonster
 
Last edited:
  • #12
24
0
The homogenous equation:
[tex]\frac{d^2y}{dx^2}+xy=0[/tex] is a negative sign off the Airy equation:
[tex]\frac{d^2y}{dx^2}-xy=0[/tex]

Therefore the solution of the original DE
[tex]\frac{d^2y}{dx^2}+xy=x^2[/tex] is given by
[tex] y = CAiryAi(-x) + DAiryBi(-x)+x[/tex]

where AiryAi and AiryBi, are independant solutions of the Airy equation.

Indeed the Airy functions are related to the Bessel functions.

Finally one can expland the answer as a series with the Gamma function appearing everywhere - nasty.
 
Last edited:

Related Threads on 2nd order DE, is there a way to solve this without series?

Replies
0
Views
2K
Replies
3
Views
963
  • Last Post
Replies
11
Views
4K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
4
Views
847
Replies
1
Views
4K
Replies
1
Views
5K
Replies
22
Views
37K
  • Last Post
Replies
0
Views
1K
Replies
1
Views
3K
Top