Variable Coefficient Differential Equation problem

[mknut389]

I have been trying to solve a differential equation that will be used to determine voltage and current variations based on ambient temperature and the distance along the line.

To avoid the complicated derrivation of the current equation... the current based off the temperature variations should be able to be solved from

I''-xI=0

when I is a function of X

I have searched all over to try and find solution to differential equations with variable coefficients, but everything I find is in a different form (and this cannot be adjusted to fit in that form).

My attempt at the solution consist of using MATLAB, MAPLE, and MATHEMATICA. All of which give crazy solutions which I do not understand.

Is there anywhere you could provide me which I could teach myself, or even better, give a description of how to solve it?

Thanks

MK

 

[ObsessiveMathsFreak]

I thought this looked familiar. Your differential equation is an http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf" . You're solutions will be in the form of Airy equations, which by the way are non elementary functions, but that's OK. It just means they're not on pocket calculators.

Technically, to obtain the solution, you can take the Fourier transform, solve a first order ODE in the Fourier domain to find a representation for the Fourier transform of y, then take the inverse Fourier transform of that representation. At this point, you can go no farther because the integral has no elementary closed form. It's used by many as the definition of the Airy function(s).

If you don't like Fourier transforms, there's always the Frobenius method... http://mathworld.wolfram.com/AiryDifferentialEquation.html" . However, I don't recommend this for human consumption.