Math 0250 Exam 1 Solutions
October 12, 1998
- Express the complex number -2+2i in polar form
- 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.
with k=0, 1, 2. In Cartesian form, these three values are
- Find complex numbers A and b so that the real function
is the real part of the complex function
F(t) = A ebt.
f(t) is the real part of
F(t) = et (1-2i)eit = (1-2i) et+it = (1-2i) e(1+i)t
- Find the inverse of the matrix
Your answer will depend on the parameter a.
- Find a basis for the solution space to the system
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
- Find all solutions (if there are any) to the system
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
- Find all solutions to the differential equation
First, divide through by t to put the equation in standard form:
The integrating factor is
and multiplying through by
This can be rewritten
and integration gives
Finally, dividing by t2 gives