MATH 310, Fall 2011: Differential Equations
Simon Fraser University
Quiz 1: Solutions
1. For the differential equation y = 4 + 3y, sketch the direction field. Hence deduce the behaviour of y(t) as t ; if this behaviour depends on the initial value y(0) at
MATH 310-3 Differential Equations Homework Set 3
Fall 2011 Some Solutions
The following solutions are not necessarily complete, but should highlight the main points for most questions. If you have any further questions regarding the problems and solutions
ASSIGNMENT 1
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
Math 303 Midterm Solutions
Instructor: Matt DeVos
First Name (please print): Last Name (please print): SFU email ID: Signature:
Problem Score Value 1 2 3 4 5 6 7 Total: 10 9 8 9 8 8 8 60
1
Problem 1. (10 points) Construct a Cayley table for the symmetry g
Homework 11
1. Let A = (0, 0) let B = (2, 1). (i) (ii) Find a line so that (B) = B = (3, 4).
Fine a line m so that m maps B to (3, 4) and A to (1, 5). is the perpendicular bisector of B = (2, 1) and B = (3, 4). We
4-1 3-2
Solution: (i): the line see that
Homework 10 Solutions
1. Use polar coordinates to prove that the transformation given by the matrix is 0, (i.e. a rotation about the origin by ). Solution: Let (r cos , r sin ) be an arbitrary point in R2 . Now we have cos - sin sin cos r cos r sin = r co
Homework 9 Solutions
1. Show that there are infinitely many plane graphs G with the following properties: (i) (ii) Every face of G is a triangle Every vertex is incident with 5 or 6 edges.
(Hint: An Icosahedron is one such graph, modify it by inserting so
Homework 7
1. Let P, be a rotation and let be any isometry. Prove that P, -1 is a rotation. Solution: Let Q = (P ) so P = -1 (Q). Then we find P, -1 (Q) = P, (P ) = (P ) = Q. So Q is a fixed point of P, -1 . Since P, is even, P, -1 (Q) is also even. Now,
Homework 6 Solutions
1. Show that if is a translation, then there exists an odd isometry so that = 2 . Solution: Let = v . Now, choose a line which is parallel to v. We claim that 2,v/2 = . which is closest To verify this, let P be an arbitrary point in R
Homework 5
1. If and m are lines which are not parallel, prove that there exists a point P and an angle so that P, ( ) = m. Solution: Let P be the unique point of intersection of angle from to m. Then P, maps and m and let be the directed to m. To see thi
Homework 4 (not for credit)
1. Suppose the lines , m, n have equations X = 2, Y = 3, and Y = 5. Find the equations for m and n m . Solution: (x, y) = (4 - x, y), m (x, y) = (x, 6 - y) and n (x, y) = (x, 10 - y). It follows that m (x, y) = m (4 - x, y) = (
ASSIGNMENT 1 SOLUTIONS
MATH 303, FALL 2011
If you find any typos or other errors please let me know. Manipulation (M1) cfw_, cfw_, cfw_, cfw_, cfw_ = cfw_, cfw_, cfw_, cfw_ = cfw_, cfw_, cfw_ (M2) There are a number of possible approaches, here is one way
ASSIGNMENT 2
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
ASSIGNMENT 2 SOLUTIONS
MATH 303, FALL 2011
If you find any errors please let me know. Manipulation (M1) (A B) = (cfw_b, d, q cfw_b, d) = (cfw_b, d, q) = cfw_a = A (M2) A - B = cfw_1, cfw_3, 8, 4 - cfw_1, 2, 3, 4, 5, 6, 7, 8 = cfw_3, 8 (M3) Yes it is an or
ASSIGNMENT 6
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
ASSIGNMENT 6 SOLUTIONS
MATH 303, FALL 2011
If you find any errors please let me know Manipulation (M1) By rule C we have that (c = d) (d = c ) c = c is valid. Then by rule F applied with A(x) being (c = x) (x - c ) and B being c = c we get that x(c = x) (
ASSIGNMENT 6
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
ASSIGNMENT 5 SOLUTIONS
MATH 303, FALL 2011
If you find any errors please let me know. Manipulation (M1) wy(x = (w, y) you could also expand out more if you want. (M2) (1 point) Which of the following are well formed formulas and which of the well formed o
ASSIGNMENT 5
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
ASSIGNMENT 4 SOLUTIONS
MATH 303, FALL 2011
If you find any errors please let me know. Manipulation (M1) No the integers with the usual are not well ordered. We can see this because the set of all integers has no least element, since for any n Z, n - 1 Z a
ASSIGNMENT 4
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
ASSIGNMENT 3
MATH 303, FALL 2011
Instructions: Do at least 3 points from each section and at least 10 points total. Up to 12 points will be graded, but your maximum score is 10. If you hand in more than 12 points please indicate which ones you want graded
Homework 3 Solutions
1. Prove that the set of all collineations in T rans(Rn ) is a subgroup of T rans(Rn ). Solution: We need to show that the collineations satisfy the identity, inverse, and closure properties (under function composition). It is immedia
Homework 2 Solutions
1. Consider the additive group Z with the operation +. Which of the following are subgroups of this group? 1. cfw_n Z : n is prime. 2. cfw_0. 3. cfw_2. 4. cfw_3n : n Z Solution: Recall that to check if a subset of a group forms a subg
Homework 1 Solutions
1. There are six symmetries of an equilateral triangle: the identity, rotate clockwise by 120 rotate counterclockwise by 120 , and three mirror symmetries about lines which pass through one vertex and the opposite edge. In class we cr
MATH 310, Fall 2011: Differential Equations
Simon Fraser University
Quiz 7: Solutions
1. Solve the differential equation y + 4y + 5y = 0, y(0) = 2, y (0) = -1.
Solution: Looking for a solution of the form y = ert , we find that the characteristic equation
MATH 310, Fall 2011: Differential Equations
Simon Fraser University
Quiz 6: Solutions
1. Consider the initial value problem y + 2y 2 = 3t + ey , y(1) = 0.
Apply one step of Euler's method with step size h = 0.01 to find an approximation y1 to y(1.01). Wr
MATH 310, Fall 2011: Differential Equations
Simon Fraser University
Quiz 5: Solutions
1. Consider the autonomous differential equation (of the form dy/dt = f (y) dy = 2y 2 (y 2 - 9). dt Sketch a graph of f (y) versus y, draw the phase line, find all the e
MATH 310, Fall 2011: Differential Equations
Simon Fraser University
Quiz 4: Solutions
1. Consider the differential equation (of Bernoulli type) t5 et t2 y = 2ty - y (t > 0, y > 0).
Show that using the substitution v = y 3/2 , this DE reduces to a linear d