I want to solve the next differential equation,

fun = t^2*x''[t] + 4/3*t*x'[t] - 2/3*(1 - (L^2)/(6*t^(2/3))) x[t] - (2*L^2)/(45*t^(2/3)) == 0

DSolve[fun, x, t]

The solutions are given in term of Bessel function. I do not know how to solve it. Any suggestion? The problem is the next enter image description here but I don't get to do.

  • 3
    Your code works for me although it takes awhile. What answer are you looking for? – Bill Watts Aug 9 at 21:20
  • 1
    I've had a little look, and by adding assumptions t>0 and L>0 and using FullSimplify I can get a version in terms of Sin and Cos. I believe that half-integer Bessel functions can be expressed in terms of trigonometric functions. You may have more luck if you supply some boundary conditions. – mikado Aug 10 at 20:42
  • 3
    If you evaluate BesselJ[5/2, L/t^(1/3)], MMA returns the result in terms of sin and cos. Having it reverse simplify back to Bessel functions might be tough. Also, I can't get your solution to satisfy your fun. Is there something missing? – Bill Watts Aug 10 at 20:49
  • 1
    The suggestion of @mikado helped to simplify the compilation time, but still does not give the solution that I am looking for and I put the real problem where it does not have initial conditions since it can be seen that it depends on constants C1 and C2. – will.al Aug 10 at 23:14
  • 2
    If you want Mathematica to prove that expr1 is equivalent to expr2, you can apply FullSimplify to expr1==expr2. For this to work, you would obviously need to find equivalent values for the integration constants C_1 and C_2 – mikado Aug 11 at 20:37

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.