I'm trying to solve a set of differential equations that all depend on a parameter, κ. I can use the system of ODEs to reduce the four equations into one second order differential equation, for y[x;κ]. I've seen certain tricks to solving equations such as d^2y/dx^2 = κ y[x].But I can not put my equation in that form. So the real trick is,how do I solve for κ and y[x] with only one equation. My idea is as follows, I will look at the limit of the differential equation at x =0 (ODE1), and the limit of it at the other boundary x = a, (ODE2).Then I will technically have two differential equations. Then I pick a value for κ, from some physics that I know about the problem, i.e. set κ = constant. After this I solve ODE1 in the domain x= 0 to x0 and I solve ODE2 in the domain x0 to x=a. Finally I check whether or not the function y[x] matches at the point x0. If it doesn't match I try this again until I find the value of κ that makes the two sides match up. Its the last part that I don't know how to implement? How do I use the mis-match to get me a new value of κ? I've read several sources about this idea, under "shooting method", but I really don't get how I should update the values of κ. Thanks for reading, and any help is much appreciated.