It is not clear what you mean by "equation". There are no true equations is your post. And it is not clear what you mean by "solve".

@ThU I used PNG; you used JPG. So I guess JPG works better here.

@WitnessA You messed up by saying soln1[n]. The square brackets mean nothing to Maple in this context. You need to make soln1 into a function of n with a command called unapply. Likewise, a(n+1) and a(n) mean nothing to Maple.

Here's my solution:

Download Airy.mw

@das1404 As far as I know, animated GIFs are always continuous. This is a limitation of GIFs, not of Maple.

@acer My results were from Maple 16.02.

@acer Can you explain the "true" results that I got?

That's very weird. Try doing a restart. You do not need to with anything to do basic arithmetic.

@tuvok1153 

First_Principle, as it's written, requires its argument to be a procedure (or appliable as a procedure); it will not (correctly) accept expressions. So, you need to use it like

First_Principle(x-> tan(x));  First_Principle(x-> sin(x)^2);

or

First_Principle(tan);  First_Principle(sin^2);

If you wish for it to work on expressions, it's a little more complicated to write:

First_Principle:= proc(f::algebraic, x::name)
local h, A:= Limit(simplify((eval(f, x= x+h)- f)/h), h= 0);
     A = value(A)
end proc:

First_Principle(tan(x), x);

Note that this requires you to pass, as the second argument, the variable with respect to which the derivative is being taken.

If you want it to work in either case, you could do this

First_Principle:= proc(f::{appliable,algebraic}, x::name)
local
     _X,
     h,
     A:= Limit(simplify(`if`(nargs=2, eval(f, x= x+h)-f, f(_X+h)-f(_X))/h), h= 0)
;
     A = value(A)
end proc:

First_Principle(tan);

First_Principle(tan(x), x);

@MikeS Your code does not work because you have MATRIX. It should be Matrix.l

Please provide an example of what you mean.

@Kitonum There's a difference between matrices being identical and them having equal entries. If you test is(A=A1), the result is false. Equal only checks that the entries are equal.

@Erfan You have misspelled the command name. It's KroneckerProduct, not KroneckeProduct.

@Hulalarip To get more information about that, you'll need to post your system of equations.

You intend to use the C variables as symbolic names, right? rather than assigning them values?

If m and m' both range from 1 to N, then m'' = m-m' will be zero or negative for many pairs (m,m'). Yet m'' is one of your function indices. How do you handle that?