Math 1b. Second Semester Calculus

Spring, 2000

Syllabus and Homework: Part III

(additional non-book problems below are also required)

April 17-21: §10.1 in Anton, pages 1-10 in the Supplement

§10.1: 2, 8, 10, 12, 16, 20, 23, 26, 28, 44, 45
Pages 9-10 in Supplement: 2, 4, 8, 9, 10, 14
A: Solve the differential equation ${dy\over dx} = y(y+1)$ .

April 24-28: Pages 10-18, page 19, pages 24-29, and pages 30-32 (up to example 3) in the Supplement

Pages 17-18 in Supplement: 2, 6, 7, 10, 12, 22, 24
Pages 28-29 in Supplement: 1, 2, 3, 4
Page 37 in Supplement: 1
A: Do the following multi-part problem:
1. Graph f(x) = (x+3)²(x-1) (justify!).

2. Assume that P = P(t) is a function of time t which satisfies

$\begin{displaymath}{dP\over dt} = (P + 3)^2(P - 1).\end{displaymath}$

3. Graph the particular solutions P(t) with P(0) = -4, -2, 2, and label all equilibrium solutions, indicating for each whether or not it is stable.

4. For which real numbers a is the solution P with initial value P(0) = a an increasing function of t?

5. By using the identity

$\begin{displaymath}{1\over (x+3)^2(x-1)} = {1\over 16(x-1)} - {x+7\over 16(x+3)^2},\end{displaymath}$

find an explicit formula for the time t in terms of the value of P(t) (i.e., given the value of P at some time, give a formula which computes what time it is). Your formula should depend upon the initial condition value P(0), which you should assume is not equal to -3 or 1. How does this formula relate to the sketches in the third part of the problem? You may not use this formula in any other part of the problem.

May 1-5: Pages 43-44 in the Supplement and review

Page 44 in Supplement: 1, 2, 3, 4
A: Let the function y satisfy the differential equation y'''=-y. Suppose that y is represented by its MacLaurin series, with infinite radius of convergence, $y=\sum_{k=0}^\infty a_k x^k$ .

1) Given that y(0)=1, y'(0)=0, and y''(0)=0, what can you say about the values of a₀, a₁, and a₂.

2) Assuming the same initial conditions as in part 1, find the general form of the coefficient a_k. (That is, write a formula for a_k in terms of k.)

3) Verify that the series with the coefficients you derived in part 2 has infinite radius of convergence.

4) Check that the function

$\begin{displaymath}y={2\over 3} e^{x\over 2} \cos \left( {\sqrt{3}\over 2} x\right) + {1\over 3} e^{-x}\end{displaymath}$

is a solution to the differential equation y'''=-y with y(0)=1, y'(0)=0, and y''(0)=0.

5) Use the uniqueness of the solution of the differntial equation y'''=-y with the given initial values in order to deduce the MacLaurin series for the function

$\begin{displaymath}f(x)={2\over 3} e^{x\over 2} \cos \left( {\sqrt{3}\over 2} x\right) + {1\over 3} e^{-x}.\end{displaymath}$

(Note that it would have been considerably more difficult to compute the entire MacLaurin series for this function directly.)

Thomas J. Brennan
2000-04-02