Thread: Z transforms problem

Please, any help with this problem on Z trabsforms:

A man's assets in connection with time k (in years) are c(k) and his expenses are e(k). They both satisfy the diff. equations c(k+1)=1,5c(k)-e(k) and e(k+1)=0,21c(k)+0,5e(k).
a. Find the % increasement of the assets per year
b. If the initial assets are 6000 Euro and the expenses during the first year are 3720 Euro, find the year whose expenses are least as well as the assets at that year's beginning.

THANK YOU VERY MUCH!

Originally Posted by jmytil

Please, any help with this problem on Z trabsforms:

A man's assets in connection with time k (in years) are c(k) and his expenses are e(k). They both satisfy the diff. equations c(k+1)=1,5c(k)-e(k) and e(k+1)=0,21c(k)+0,5e(k).
a. Find the % increasement of the assets per year

We have:

c(k+1) = 1.5 c(k) - e(k)
e(k+1) = 0.21 c(k) + 0.5 e(k).

where k=0 denotes the initial conditions

The percentage increase in k-th year c is:

[c(k)-c(k-1)]/c(k-1) *100 % = [1.5 c(k-1) - e(k-1) - c(k-1)]/c(k-1) *100%

simplifying we get the percentage change in the k-th year is:

50 -e(k-1)*100/c(k-1) %

b. If the initial assets are 6000 Euro and the expenses during the first year are 3720 Euro, find the year whose expenses are least as well as the assets at that year's beginning.

I'm afraid I don't understand what part b. is asking for.

RonL

Thanks yu very much!
The teacher said that we must use Z transforms and that question (a) gives a result 20%.
I think that in the second question he gives us that c(0)=6000 and e(0)=3720 and asks for the value of k where e(k) is minimum and for that k, the value of c(k-1).

Originally Posted by jmytil

Thanks yu very much!
The teacher said that we must use Z transforms and that question (a) gives a result 20%.

Then you are likely not asking us the question as you have been asked it.
It seems possible you are asking for the asymtotic % increment in assets
per year as k -> infty.

RonL

We have:

c(k+1) = 1.5 c(k) - e(k)
e(k+1) = 0.21 c(k) + 0.5 e(k).

where k=0 denotes the initial conditions

This is a second order difference equation, so lets make it explicit.

Rearrange the first equation so:

e(k) = 1.5 c(k) - c(k+1) ... (A)

and :

e(k+1) = 1.5 c(k+1) -c(k+2) ... (B)

Now substitute (A) and (B) into the second equation:

1.5 c(k+1) -c(k+2) = 0.21 c(k) + 0.5 [1.5 c(k) - c(k+1)]

Rearrange:

c(k+2) - 2 c(k+1) + 0.96 c(k) = 0.

Which is a linear costant coefficient differenece equation.

Now take the z-transform of this to give:

z^2 C(z) - z^2 c(0) -z c(1) -2 z C(z) - 2 c(0) +0.96 C(z) = 0,

or:

C(z)[1-2 z^-1 +0.96 z^-2] = [c(0) - z^-1 c(1) -2 z^-2 c(0)]

C(z) = [c(0) - z^-1 c(1) -2 z^-2 c(0)]/[1-2 z^-1 +0.96 z^-2] ... (C)

So to find c(k) in tems of c(0) and c(1) you need to invert the RHS of (C),
then the asymtotic year on year % increase on assets is:

lim(k-> infty) 100 (c(k+1)-c(k))/c(k).

RonL