1550078481-Ordinary_Differential_Equations__Roberts_

540 Ordinary Differential Equations

Exercises 1.3 Solutions and Problems

7. a. y = 2x ln x / ln 2

b. No unique solution. There are an infinite number of solutions of

the form y = c 1 x lnx where c 1 is arbitrary.

c. No solution.

9. y = 2x - x^2 + x^3

Exercises 1.4 A Nobel Prize Winning Application

l. 59.53; 35.83; 675.7 years 3. 133% grams; 2.4 years

Chapter 2 THE INITIAL VALUE PROBLEM:

y' = f(x, y); y(c) = d

Exercises 2.1 Direction Fields

l. A<--> b, B <-->a 3. E <--> e, F <--> f 5. I <--> j, J <--> i

7. The direction field is defined in the entire xy-plane. The function
   y = 0 is a solution on ( -oo, oo). In the first and third quadrants, the
   solutions are strictly increasing and in the second and fourth quadrants,
   the solutions are strictly decreasing. Thus, relative minima occur on the
   positive y-axis and relative maxima occur on the negative y-axis. If a
   solution is positive for x > 0, then y(x) ---+ +oo as x ---+ +oo. If a so lu-
   tion is negative for x > 0, then y(x) ---+ -oo as x ---+ +oo. If a solution
   is positive for x < 0, then y(x) ---+ +oo as x ---+ -oo. If a solution is

negative for x < 0, then y(x)---+ -oo as x---+ -oo.

9. The direction field is undefined on the y-axis where x = 0. The function

y = 0 is a solution for x =f. 0. In the first and third quadrants the

solutions are strictly increasing and in the second and fourth quadrants

the solutions are strictly decreasing. If a solution is positive for x > 0,