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

[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).

 

[Allah]

can you use integrating factor?

 

[aeroegnr]

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

 

[Dr Transport]

The homogeneous equation looks like the Airy equation...

 

[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.

 

[arildno]

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

 

[noppakhuns]

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

 

[Dr Transport]

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

 

[Max0526]

Hi;
Look at this: http://mathworld.wolfram.com/AiryDifferentialEquation.html.
Max.

 

[selfAdjoint]

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).

 

[cookiemonster]

Mathematica gives:

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

cookiemonster

 

[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.