Math 246: Recommended Problems from the Textbook (9th edition)

These problems will be similar to problems of the exams and the final exam. But they will not be collected or graded. Note that partial solutions to the problems are in the back of the textbook.

Section 1.1 (Introduction), p.7

For the differential equations 1-4 do the following:
(i) try to sketch the direction field by hand. Based on the direction field: determine the behavior of y(t) for t going to infinity depending on the initial value at t=0
(ii) find the general solution of the differential equation using the hint that the solution is of the form y(t) = a + C*exp(r*t).
(iii) find the solution of the initial value problem with initial condition y(0)=1.

Section 2.1 (Linear Equations), p. 39

only find the general solution: 1, 2, 6, 10
13, 14, 16, 18

Section 2.2 (Separable Equations), p. 47

1,2,3,4,9,10,13,21

Section 2.3 (Modeling with First Order Equations), p. 59:

1,3,5, 28 (cf. Ex.1 on p.2), 29 (cf. Ex.4 on p.58)

Section 2.4 (Differences between Linear and Nonlinear Equations, p. 75

1, 2, 3, 14, 15

Section 2.5 (Autonomous Equations and Population Dynamics), p. 88

3,4,5,9,10,11,20

Section 2.6 (Exact Equations and Integrating Factors), p. 99

1,2,3,7,10,13,19. For which of those problems can you find the solution in explicit form?

Section 2.7, 8.1, 8.2, 8.3, 8.6

Problems for Numerical Methods

Section 3.1 (Homogeneous Equations with Constant Coefficients), p. 144

1,5,8,9,10

Section 3.3 (Complex Roots), p.163

7,9,12,17,21

Section 3.4 (Repeated Roots), p. 171

1,11,12,13,14

Section 3.5 (Method of Undetermined Coefficients), p. 183

1, 3, 6,7; only (a) for: 19,20,23

Section 3.6 (Variation of Parameters), p. 189

5,7,10

Section 3.7 (Mechanical and Electrical Vibrations), p. 202

1, 2, 5, 11, 17
Note: For some strange reason the book does not tell you directly what the spring constant k is. The weight W is the force with which mass is pulled by earth's gravity. It is related to the mass M by W = M g , where g = 9.81 m s^-2 = 32 ft s^-2.

• "A spring is stretched a distance of D by a force of F"
  means that the spring constant is k = F/D.
• "A mass weighing W stretches a spring a distance D"
  means that the spring constant is k = W/D.
• "A mass of M stretches a spring a distance D"
  means that the spring constant is k = Mg/D.

For the exams I will only give you problems without units, and I will tell you directly what the spring constant k is.

Section 3.8 (Forced Vibrations), p. 215

1, 9, 12, 17 (use formulas (14), (15) on p.209 for (d))

Section 6.2 (Solution of IVP using the Laplace Transform), p. 320

11, 12, 14, 21, 22, 23

Section 6.3 (Step Functions), p. 328

1, 2, 7, 10, 20, 21, 24

Section 6.4 (Discontinuous Forcing Functions), p. 336

1, 5, 9, 10

Section 7.5 (Homogeneous Systems of First Order Linear Equation), p. 398

Find the general solution, find the type and stability of the critical point (0,0), sketch the phase portrait by hand: 2, 3, 4, 5

Section 7.6 (Complex Eigenvalues), p. 409

Find the general solution, find the type and stability of the critical point (0,0), sketch the phase portrait by hand: 1, 3, 4, 6

Section 7.7 (Fundamental Matrix), p. 420

only (b): Find exp(At) which is fundamental matrix with Φ(0)=I: 1, 3, 4

Section 7.8 (Repeated Eigenvalues), p. 428

Find the general solution, find the type and stability of the critical point (0,0), sketch the phase portrait by hand: 1, 2

Section 9.1 (The Phase Plane: Linear Systems), p. 495

(do parts (a), (b), (c) in all problems): 5, 6, 10, 12

Section 9.2 (Autonomous systms), p. 506

only (a): 18, 19, 21

Section 9.3 (Nonlinear Systems), p. 516

do parts (a), (b), (c) in all problems: sketch a phase portrait close to each critical point, try to sketch a plausible global phase portrait: 5, 6, 7, 8, 10, 12