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DSolve:: Bessel's differential equation
Hi buddies, Here is a differential equation , which could not be done by DSolve (in Version 5.0 ). This occurs in the mathematical modelling of a simple pendulum of length L and a parameter k . (Actually , k ^ 2 = frequency of oscillation^2 / acceleration due to gravity ) (L - x) y''[x] - y'[x] + k^2 y[x] == 0 .... (1) The above equation is in fact reducible to Bessel's differential equation (with order n = 0 ) with the substituions L-x = z and s = 2 k Sqrt[z] y''[s] + 1 /s y'[s] + y[s] ==0 .... (2) The text-book says that the solution of eqn (1) contains a BesselJ[0,2 k Sqrt[L-x] ]. How can I get DSolve to answer (1) directly , without resorting to eqn (2)?