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

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).

Related Differential Equations News on Phys.org
can you use integrating factor?

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

Dr Transport
Gold Member
The homogeneous equation looks like the Airy equation.....

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
Homework Helper
Gold Member
Dearly Missed
You could say that you get a "nice" general solution by dubbing it as Ai(x)..

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

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

Staff Emeritus
Gold Member
Dearly Missed
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).

Mathematica gives:

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

Last edited:
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: