Y''=-(t^2)y differential equation

[HomogenousCow]

Hello I was recently working on a problem where I had to solve the differential equation in the title ( where y is a function of t), I found an exact series solution through peturbation theory in which a pattern emerged between successive orders.
However, the series solution is not very useful for computation and I would like to ask if anyone knew a special function which solved this equation?

 

[hilbert2]

That's a special case of the Weber differential equation, and the solution can be represented with parabolic cylinder functions, see:

http://en.wikipedia.org/wiki/Weber–Hermite_function
http://mathworld.wolfram.com/WeberDifferentialEquations.html

 

[the_wolfman]

You can also use the change of variable t^2=x, the resulting differential equations is a modified Bessel equation for x whose solutions are modified Bessel Functions.