Name______
Math 44 – Differential Equations
Friday, February 9, 2007
Graded Problem Set 1
Due at start of class Wednesday, February 14, 2007. This problem set substitutes in some measure for an exam. You may not consult with anyone other than the instructor, but you may use references (including the text and your notes), calculators or computers, and any software. (There are 9 numbered problems on 3 pages.)
Analytic Techniques
1. Find all of the solutions (on any interval) of .
2. Find all of the solutions (on any interval) of .
3. Find any one solution, on any interval, to .
4. (Beware, tricky) One solution to that is valid for all t is the
zero function, y1(t) = 0 for all t.
Find another solution that is also valid for all t.
5. If y(0) = 0 and whenever y > –1 , what is y(3) ?
Numerical Techniques
6. Consider again our old friend, with y(0) = 0.
Estimate y(1) to an accuracy of 0.001. You may use software. You may use Euler’s method or any other method, but please don’t mention Bessel functions.
Qualitative Techniques
7. Here’s a graph of (y – y3), as a function of y
Note that (y – y3) = 0 when y = 0, +1, or –1.
Note that ety is always positive.
Consider the equation .
a. Make a reasonable drawing of the direction field. (Use software if you like. You can
re-draw the direction field by hand or, if you can get your software to print, turn
in a printed image.)
b. In the drawing, show three separate solutions y1, y2, y3, with
y1(-1) = ½, y2(0) = ½, y3(+1) = ½.
c. Estimate y2(100) to an accuracy of 0.000000000000000000000000001.
(No proof required. Don’t use a calculator or computer for this part.)
Substitution
8. Here is “Bernoulli’s equation:”
.
It would be a linear equation, if only that pesky factor yn were missing from
thelast term. (Let’s take n to be any integer constant other than 0 or 1.)
a. If the functions z and y are related by for every t, (where k
is a constant to be named later) thenwhat is y in terms of z?
y(t) = ______(Careful --- NOT z-k.)
b. What is in terms of z and ?
= ______
c. Using the results of (a) and (b), restate the original equation in terms of z’, z, and t.
(y should not appear.)
= ______
d. What must k be in order for the answer to part (c) to be a linear equation?
e. Use this approach to find any one solution (on any interval) to .
An Application
9. The Dresden speaker (Trachette Jackson) gave the following equation for the number of tumor cells in a non-vascular tumor, N(t), as a function of time t:
.
In this equation, N(t) is an unknown function defined and continuous for t ≥ 0,
and r1, r2, c1, c2, and C are all known constants.
In the model, N(0) is assumed to be a very small positive number.
a. Using the results of problem 8 OR the techniques from the example(s) at the end of Section 1.1, find the general solution to this equation (at least for N > 0).
b. Draw a rough graph of a particular solution for which N(0) is a small positive number.
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