Consider the problem of calculating the Laplace transform of

Consider the problem of calculating the Laplace transform of a periodic function. Compute the Laplace transform of the saw tooth function z(t) = t - [t] where t denotes the greatest integer less than or equal to t. (See Exercise 18 in Section 6.3) Consider the initial value problem dy/dt = -y + z(t), y(0) = 0 where z(t) is the saw tooth wave above. Use your work above to compute the solution to this I VP by hand, and then compare your answer with what Maple gives. Plot the resulting solution.

Solution

Given that Z(t)=t-[t]

[t] is denotes less than or equal to t

let consider [t] is less than t

then z(t)=t-n where 0<n<t

laplase transform of z(t) is

L{z(t)}=L(t-n)

   =L(t)-L(n)

   =L(t)-nL(1)

=(1/p²)-n(1/p) where p>0

=(1-np)/p²

now consider [t]=t then

z(t)=t-t

=0

then

L{z(t)}=L(0)

L(0) is not defind.