On rk4 of fortran

Your problem is that k is undefined, so it is considered as an integer, and dydx in diff_equation is undefined, so it is considered as a floating point number. Inconsistent types lead to strange numbers.

I strongly recommend that you add:

Implicit none

Then for each array of each program, the variable declares its type.

Don't use dimensions, and forget the word forever.

An integer array is defined as follows:

Integer:: i(n)

Floating-point number groups are defined as follows:

Real number:: r(n)