Problems due Wednesday 3 Apr 2024 by 11:59pm
Easier Problems
Problem 11-01
Laplace transforms
Find the Laplace transforms of the following functions:
- (a)
(do the derivation, don’t just cite the chart) - (b)
(hint: trig identities)
Problem 11-02
Inverse Laplace transforms
Find the inverse Laplace transforms of the following functions:
- (a)
- (b)
- (c)
Harder Problems
Problem 11-03
Periodic functions
Suppose that
is periodic with period . This means that satisfies the equation for any . Notice that this equation implies:
So the value of
at any point can be traced back to the values in the range . Show that for such
, its Laplace transform is given by:
Problem 11-04
Ramp drive
A ramp function is given by the piecewise function
defined for some : This function may be written using the Heaviside step function as follows:
.
- (a) Find the Laplace transform
of the step function. - (b) Relate the result of (a) to the shift operation from lecture:
. - (c) Find the Laplace transform
of the ramp function using the result of (a). - (d) Now solve the differential equation:
with . - (e) [extra credit] Find the Laplace transform of the sawtooth function which is a sequence of consecutive ramps:
for and . (Hint: Use Problem 11-03.)