Math 0250 Exam 1 Solutions

October 12, 1998

F. Beatrous

1.
(a)
Express the complex number -2+2i in polar form .

Solution:

(b)
Find all cube roots of -2+2i. You may leave your answer in polar form if you wish, but you should include each cube root exactly once in your list of answers.

Solution:
Write

and

with k=0, 1, 2. In Cartesian form, these three values are

and

2.
Find complex numbers A and b so that the real function is the real part of the complex function F(t) = A ebt.

Solution:
f(t) is the real part of

F(t) = et (1-2i)eit = (1-2i) et+it = (1-2i) e(1+i)t

3.
Find the inverse of the matrix

Solution:

so

4.
(a)
Find a basis for the solution space to the system

Solution:
The augmented matrix

row reduces to

which yields the equivalent system

This system can be solved for x1 and x3 in terms of x2 and x4. Assigning x2 and x4 the arbitrary values s and t respectively gives

In vector notation, this can be written

A basis for the space of solutions is therefore

(b)
Find all solutions (if there are any) to the system

Solution:
You can come up with a single solution by trial and error. This one will do:

Combining this with the general solution to the associated homogeneous system found in part 4a give the general solution

5.
Find all solutions to the differential equation

Solution:
First, divide through by t to put the equation in standard form:

The integrating factor is

and multiplying through by gives

This can be rewritten

and integration gives

Finally, dividing by t2 gives

Frank Beatrous
1998-10-13