@Carl Love 

Had a read and attempted again and think I've got it now. Does this look right?

invztrans(z*(5*z/(5-z))^2/(z-2), z, n);
125 n 100 n 125 n
--- 5 + --- 2 + --- 5 n
9 9 3
sum(f*2^(n-k)*k5^k, i = k .. n);
(n - k) k
(n - k + 1) f 2 k5
simplify(((125/9)*5^n+(100/9)*2^n+(125/3)*5^n*n)/((n-k+1)*f*2^(n-k)*k5^k));
(-k) / (-n + k) n (-n + k) n k\
25 k5 \15 2 5 n + 5 2 5 + 4 2 /
- -----------------------------------------------------
9 (-n + k - 1) f

@nm

 Using what it says on the help page but the command isn't giving a solution. Should probably add that I am very new to using maple and have been thrown in the deep end with this. Apologies if I sound vague

invztrans(f, z, n);
f charfcn[0](n)
invstrans(z/(z-2), 5*z/(5-z), y, n);
/ z 5 z \
invstrans|-----, -----, y, n|
\z - 2 5 - z /

@Preben Alsholm 

Thanks for the help! I believe that is the way I needed to solve the equations. I just need to try to apply it to the spring equations applying newtons 2nd law to the masses.

md^2x/dt^2=-kx-k(x-y)

md^2y/dt^2=-k(y-x)-ky

when x and y are the displacement from the equilibrium. Initially the masses are displaced from their equilibrium posistions and released so that

x(0)=a, y(0)=b, Dy(0)=Dx(0)=0

assuming k1=1

it simplifies to m d^2x1/dt^2=-2x-y and

d^2y/dt^2=-2y-x

using these equations I have to use the solved laplace equations to solve them

@RAfossey 

restart;with (inttrans):
a:=1;b:=1;c:=1;d:=1;x(0):=0;Dx(0):=0;y(0):=0;Dy(0):=1;
1
1
1
1
0
0
0
1
f:=t->x+y;
t -> x + y
eqt:=diff(diff(y(t),t),t)=a*x(t)+b*y(t); eqt2:=diff(diff(y(t),t),t)=c*x(t)+d*y(t);
d / d \
--- |--- y(t)| = x(t) + y(t)
dt \ dt /
d / d \
--- |--- y(t)| = x(t) + y(t)
dt \ dt /
eqt:=laplace(eqt,t,s); eqt2:=laplace(eqt,t,s);
2
s laplace(y(t), t, s) - D(y)(0)
-------------------------------- =
2
s

laplace(x(t), t, s) + laplace(y(t), t, s)
-----------------------------------------
2
s
2
s laplace(y(t), t, s) - D(y)(0)
-------------------------------- =
3
s

laplace(x(t), t, s) + laplace(y(t), t, s)
-----------------------------------------
3
s
eqt:=subs(y(0)=0,D(y)(0)=1,x(0)=0,D(x)(0)=0,laplace(x(t),t,s)=xbar,laplace(y(t),t,s)=ybar,eqt); eqt2:=subs(y(0)=0,D(y)(0)=1,x(0)=0,D(x)(0)=0,laplace(x(t),t,s)=xbar,laplace(y(t),t,s)=ybar,eqt);
2
s (xbar + 1) xbar + 1
------------- - 1 xbar + --------
2 2
s - 1 s - 1
----------------- = ---------------
2 2
s s
ybar:=solve(eqt,ybar);
xbar + 1
--------
2
s - 1

UPDATE- this is as far as I got with the code but am struggling with correctly applying the correct values, and applying it to the mass spring

@Preben Alsholm 

Ah ok, I think I understand how to solve the solutions, the problem being that I need to use techniques of Laplace transforms to solve. I know that start involes using (inttrans) and specifiying the values.

Then using laplace(eqt,t,s); and substituting values to complete the code. 

Problem being I am very unfamiliar with Maple and the commands are causing me problems about how to complete