I define the following functions:

left[x_] := -c Exp[alpha x]

box[x_] := a Sin[k x] + b Cos[k x]

right[x_] := c Exp[-alpha x]

dleft[x_] = D[left[x], x]

dbox[x_] = D[box[x], x]

dright[x_] = D[right[x], x]


Next I have a system of 4 equations:

Eliminate[{

left[-l] == box[-l] &&

dbox[l] == dright[l] &&

right[l] == box[l] &&

dleft[-l] == dbox[-l]
}, {a, b, c}, InverseFunctions -> True]


I expect to get k cot[k l] = -alpha. Instead I get "True" as the answer.

IF I use the command (drop c from elimination list):

Eliminate[{left[-l] == box[-l] && dbox[l] == dright[l] &&
right[l] == box[l] && dleft[-l] == dbox[-l]
}, {a, b}, InverseFunctions -> True]


I get: alpha c Sin[k l] == -c k Cos[k l] which is the correct answer if you cancel c from both sides.

Can you please tell me what am I doing wrong ?

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Eliminate uses the given set of equations to eliminate the requested variables from those equations, returning the equations not needed to do the eliminations. Thus,

Eliminate[{left[-l] == box[-l] && dbox[l] == dright[l] && right[l] == box[l] &&
dleft[-l] == dbox[-l]}, {a, b}, InverseFunctions -> True]


eliminates variable a and b, returning the remaining equation.

(* alpha c Sin[k l] == -c k Cos[k l] *)


To Eliminate the remaining variable, c requires the use of the remaining equation

Eliminate[%, {c}]


leaves no equations, and Eliminate returns

(* True *)


(Why it does not return Null, instead, I do not know.)

This behavior is illustrated perhaps more clearly in the simple example,

Eliminate[e == 1 && d == 1, e]
(* d == 1 *)
Eliminate[%, d]
(* True *)