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

1. Jun 17, 2004

### aeroegnr

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).

2. Jun 18, 2004

### Allah

can you use integrating factor?

3. Jun 18, 2004

### aeroegnr

For a 2nd order equation? I know how to do that for first order but not second order equations.

4. Jun 18, 2004

### Dr Transport

The homogeneous equation looks like the Airy equation.....

5. Jun 18, 2004

### aeroegnr

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
$$\int e^x^2 dx$$

Which of course can only be done with power series, and thought I could integrate it and give a nice general solution.

6. Jun 19, 2004

### arildno

You could say that you get a "nice" general solution by dubbing it as Ai(x)..

7. Jun 19, 2004

### noppakhuns

If y1 is known, you can use reduction of order to solve this.

8. Jun 19, 2004

### Dr Transport

if memory serves me correctly, the Airy equation is proportional to a Bessel function of 1/3 order......Look out there online.

9. Jun 22, 2004

### Max0526

10. Jun 24, 2004

Staff Emeritus
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. Jun 24, 2004

Mathematica gives:

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

Last edited: Jun 24, 2004
12. Jun 30, 2004

### heardie

The homogenous equation:
$$\frac{d^2y}{dx^2}+xy=0$$ is a negative sign off the Airy equation:
$$\frac{d^2y}{dx^2}-xy=0$$

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

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: Jun 30, 2004