# Solve differential equation [closed]

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 but I don't get to do.

## closed as off-topic by Mariusz Iwaniuk, MarcoB, bbgodfrey, gpap, ÖskåAug 24 '18 at 8:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – Mariusz Iwaniuk, gpap, Öskå
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• Your code works for me although it takes awhile. What answer are you looking for? – Bill Watts Aug 9 '18 at 21:20
• 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 '18 at 20:42
• 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 '18 at 20:49
• 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 '18 at 23:14
• 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 '18 at 20:37