How to set up a system of linear equations with loop and solve it?

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There is a system of linear equations with 1000 equations and 1000 variables. The 1000 variables are: x(1),x(2),...,x(999),b. The 1000 equations are: 6*x(i)+(1000-i)*(x(i+1)-x(i))-5*x(i+500)=1 (i=0,1,...,499), and x(i)=1+b*(2/3-i*2/3000)^6 (i=500,501,...,999), where x(0)=0.8. How to program this system of linear equations and solve it?
Torsten on 18 May 2018
Edited: Torsten on 18 May 2018
If you understood the solution of the two ODEs that are coupled at x=1/3, you should also come up with a solution of this linear system of equations that is coupled at i=500.
The "real" unknown is b which can be determined by making x(500) from the first recursion equal x(500) from the second recursion.
Best wishes

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