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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.