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.

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    Your code works for me although it takes awhile. What answer are you looking for? – Bill Watts Aug 9 at 21:20
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    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
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    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
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    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
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    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

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