# Use Solve[] with Bessel, gamma, and hypergeometric functions?

I need to find values of {a,b,c} such that the 0th, 2nd, and 4th order moments of f[x]=Exp[-ax^4 - bx^2 - c] will equal respectively {1,2,10}.

I didn't really expect this to work, and it didn't:

Solve[{
Integrate[Exp[-a*x^4 - b*x^2 - c], {x, -Infinity, Infinity}] == 1,
Integrate[(x^2)*Exp[-a*x^4 - b*x^2 - c], {x, -Infinity, Infinity}] == 2,
Integrate[(x^4)*Exp[-a*x^4 - b*x^2 - c], {x, -Infinity, Infinity}] == 10},{a, b, c}]

Is there any clever way I can use Mathematica to solve this problem without having to write my own custom iterative solver? (I will settle for a non-clever way.)

• Does {a -> 0.0108063, b -> 0.141937, c -> 1.36499} float your boat? If so, set it up for use of FindRoot...
– ciao
Jul 28, 2015 at 6:06
• Solve[] is really only intended for algebraics and "simple" transcendentals. Things as complicated as yours require the use of FindRoot[] along with some amount of numerical/analytic insight on where the roots might be. Jul 28, 2015 at 6:11
• In any case… Jul 28, 2015 at 6:13
• @J. M.: Seriously nice answer ref'd there, +1 there!
– ciao
Jul 28, 2015 at 6:21
• @ciao, thanks, but Maxim and Daniel really should get the lions' share of the credit; I wouldn't have been able to come up with that if Daniel hadn't answered my other question. :) Jul 28, 2015 at 6:24