Can any one help me to solve a differential equation of the form
d2y/dx2 + f(x) y = 0
where, f(x) can be any kind of function
e.g. f(x)=sin2x or exp(x) or ax2+bx+c or cosech(x) etc.
ODE's without constant coefficients are hard to find general solutions to.
One common method is to attempt to find a power series solution centered around your initial conditions. (The other is numerically)
It usually goes something like this suppose that
Note that I am using the to be the constants in the series expansion
Now we need to do the same thing with this gives
Now comes the bad part we need to product of the series
Using the columns gives the series representation
Now just equate coefficients
This gives
Now you should know all of the and the first two your initial conditions and this will give a recurrence relation for all of the coefficients of the power series.