Math 432 – HW 7.2 Solutions

__Assigned:__ 2, 5, 7, 9, 12,
13, 17, 20, 21, 23, 24, 27, 29(a, c, e, g, and i),
and 30

__Selected
for grading:__
2, 13, 20, 24

__Solutions.__

2. The Laplace transform of *f* (*t*) = *t*^{
2}.

6. The Laplace transform of *f* (*t*) = cos *t*:

To find the indefinite
integral , you could use the
integral table at the front of the book, or you could do the following.

First, you use
integration by parts:

*u* = cos
*t*,
*du* = –sin *t* *dt*, *dv* = *e*^{ –st} *dt*,

And you use integration
by parts again:

*u* = sin *t*,
*du* = cos
*t* *dt*, *dv* = *e*^{ –st} *dt*,

Adding that last term to
both sides of the equation gives

And so

Taking the limit of the
definite integral gives

7. The Laplace transform of *f* (*t*) = *e*^{
3t} cos
3*t* is

A substitution similar
to the one made in #5 yields

And the limit of this
expression as *N* → ∞ is

9. The given function is defined piecewise by

The Laplace
transform is

12. The function whose Laplace transform we want is
given by

The Laplace transform
is

13.

And the domain for this
function is the smallest interval common to all four domains. That's
*s* > 0.

17.

20.

21. Here is the sketch of the function given by

Since each
formula in the definition is continuous on its respective interval of
definition, the only question is whether
*f* is continuous at *t* =
1. As you can check, *f* (*t*)
→ 1 = *f* (1) as *t* → 1, so *f* is continuous on the interval [0, 10].

23. The function given by

is not continuous on
[0, 10] since it is not continuous at *t* = 1
and not continuous at *t* = 3.

The one-sided limits and both exist and are finite.

The one-sided limits and both exist and are finite.

So this function is
piecewise continuous on [0, 10].

Here is its graph.

24. The function

is not continuous
at *t*
= 2 and not continuous at *t* =
–2 (but this second value is outside the range of our consideration).

The one-sided limits are both finite.

So this function is
piecewise continuous on [0, 10].

Here is its graph.

27. And here's one more.

This function is
neither continuous not piecewise continuous on [0, 10] because the limit is not finite.
Here is its graph.

29. (a) *t*^{ 3} sin *t* is of exponential order
α = 1. (In fact, any *α* > 0 would do.)

(c) No matter how large the constants α and *M*, the absolute value will eventually (i.e., for large enough *t*)
be larger than *Me*^{ αt}, so
this function is not of exponential order.

(e) Since for large
*t*, then this function is not
of exponential order.

(g) sin(*t*^{
2}) + *t*^{ 4}*e*^{ 6t} is of exponential
order α = 6.

(i) for all
*t* > 0 and

So this function is of
order α = 2.

30. For all the transforms in Table 7.1,