Math 308 - Spring 2001
Homework
Assignment 1
- Read sections 1.1-1.3.
- Solve:
- y' + y = t et (Ex. 4, § 1.2)
- y' - 2ty=1, y(0) = 1 (Ex. 14, § 1.2)
- (1+t2)y' + 4ty=t, y(1) = 1/4. (Ex. 16, §
1.2)
- The velocity in the parachute problem that we discussed in class
satsified the ODE v'= -g - (k/m)v. Recall that the limiting velocity
for the special case where v(0)=0 was vL = - mg/k. Show
that no matter what v(0) is, the velocity v(t) has the same limit,
vL, when t -> infinity.
We will discuss these on Thursday 18 January.
Assignment 2
- Read sections 1.4, 1.5, 1.8.
- Do these problems:
- §1.3: 1, 3, 7.
- §1.4: 1, 6. Plot the solution to 6.
Due: 25 January.
Assignment 3
- Read sections 1.9, 1.10 (pgs. 67-71, 76-79), 1.13
- Do these problems:
- §1.5: 2, 8 (Plot the solution.), 10.
- §1.7: 7 (Plot V(t).), 11.
Due: 1 February.
Assignment 4
- Read sections 1.10 (pgs. 76-79), 1.13.
- Do these problems:
- §1.8: 7 (Plot the solution.), 10.
- §1.9: 4, 9, 13.
Due: 8 February. Also, Project 1 is
due 13 February.
Assignment 5
- Read sections 2.1 and 2.2.
- Do these problems. For 2 and 3 below, use stepsizes 0.1, 0.05,
0.025, 0.00125. Do the exercises with MATLAB or MAPLE. You may use
eulr or rk308h. Plot the errors.
- §1.10: 16
- §1.13 (p. 100): 5.
- §1.16: 5.
Due: 15 February. Also, Project 1 is
due 13 February.
Assignment 6
- Read section 2.2.
- Do these problems.
- §2.1: 2(c,d,f), 5, 8, 11
- Use the following method to find a second solution to
y''+2y'+y=0, given that y1 = e-t is a known
solution to the equation. Solve W'+pW=0 to find the Wronskian
of any two solutions; show that y2 satisfies the first
order equation
y'2 +y2 = et W(t)
Solve this equation to obtain y2.
Due: 22 February.
Assignment 7
- Read sections 2.3 and 2.4.
- Do these problems.
- §2.2 (pg. 140): 4, 6, 10
- §2.2.1 (pg. 144): 2, 5 (plot y(t)), 15
- §2.2.2 (pg. 149): 12
- Find the polar form for 1-2i, 3+4i, and 1/(1+i).
- Plot z(t) = exp(-(1+4i)t) for t=0 to t=pi. (This is a curve in
the complex plane.)
- Find the polar form of all six solutions to z6=1. Plot
them in the complex plane.
Due: 1 March.
Assignment 8
- Read section 2.5.
- Do these problems.
- §2.3: 3, 5.
- §2.4: 1, 5.
Due: 8 March.
Assignment 9
- Read section 2.6.
- Do these problems.
- §2.5: 2, 7, 11, 14, 17, 18.
- in problem 7 above, solve the IVP y(0)=1, y'(0)=2; plot the
solution for t=0 to t=4.
Due: 22 March.
Assignment 10
- Read sections 2.9 and 2.10.
- Do these problems.
- §2.6 (pg. 172): 1, 5 (plot the solution), 6 (Hint:
cos3(x) = (3/4)*cos(x) + (1/4)*cos(3x).), 9, 13
- §2.6.2 (pg. 177): 1, 2(b) (plot the solution), 6
Due: 29 March.
Assignment 11
- Read sections 2.11-2.13 and 3.1.
- Do these problems.
- §2.9 (pgs. 232-233): 7, 9, 23.
- §2.10 (pgs. 237-238): 6, 7(a), 14, 15, 19, 23.
- §2.11 (pg. 243): Find Y(s) in exercise 9. (Don't invert the
transform, yet.)
Due: 5 April.
Assignment 12
- Read sections 3.1, 3.8, and 3.9.
- Do these problems.
- §2.11 (pg. 243): 7, finish 9.
- §2.12 (pgs. 250-251): 4, 7, 8(b) (Hint: interchange the
Laplace transform and infinite sum).
- §2.13 (pgs. 256-257): 3, 12, 13.
Due: 12 April.
Assignment 13
- Read sections 3.8 and 3.9.
- Do these problems.
- §3.1 (pgs. 271-273): 9, 12, 15.
- §3.8 (pgs. 340-341): 1, 3, 7, 8.
Due: 19 April.