In the system below, I need to solve the solution algebraically (it is known in advance that from "V3" that a0=1,just open the V3 command).

V := exp(lambda*S) = S^4*a4 + S^3*a3 + S^2*a2 + S*a1 + a0;
V1 := subs(S = 2, V);
V2 := subs(S = 1, V);
V3 := subs(S = 0, V);
V4 := subs(S = -1, V);
V5 := subs(S = -2, V);
fsolve(subs(a0 = 1, {V1, V2, V4, V5}), {a1, a2, a3, a4});

I already know the answers, but I need maple to provide me with the command in the form

a1:=(1/6)*[8*sinh(lambda)-sinh(2*lambda)] and

a2:=(1/12)*[16*cosh(lambda)-cosh(2*lambda)-15],

a3:= ... etc.

What is the best way to do this?

Once the terms have been selected, as below

I only need to keep terms that contain a certain function with generic arguments, for example, tanh(x), in this case, that is, I want from A01

-(m2/4) tanh(2k-2ç)+(m2/16) tanh(4k+2ç)+-(m2/4) tanh(2k+2ç)+(3m2/8) tanh(2ç)

and delete the terms

3/16-(3/16)m2^2

What's the best command to make that? In other words, always exclude terms that do not accompany tanh

For any given function solely and exclusively given by exponentials, as follows (expression A1), I need to select the argument of each exponential and apply it to any function, for example COS(X)

A1 :=  +5

That is, we would have

A1':=+5

I know this result is zero, but this is a simple example. What is the best way to do this?

How to apply the following relationship in maple commands? In this physics this relation it's utilized in effective field theory

I need some command that simplifies the exponentials (or a series of commands) so that there is no power over power.

What's happening (exp(k))^2

What do I need exp(2k)

What would be the most appropriate method? Below is a photo of my code