DSolve can find solutions for most of the standard linear second-order ODEs that occur in applied mathematics.
Here is a plot that shows the oscillatory behavior of the Airy functions for large negative values of .
The solution to this equation is given in terms of the derivatives of the Airy functions, AiryAiPrime and AiryBiPrime.
Here is the solution for Bessel’s equation with . Note that the solution is given in terms of Bessel functions of the first kind, BesselJ, as well as those of the second kind, BesselY.
Here is a plot of the BesselJ functions for specific values of .
With set to 5, the solution is given in terms of polynomials, exponentials, and Erfi.