MATH2860U Midterm
May 26, 2005
1
1. (1 mark each; total 6 marks) In the questions below, CLEARLY circle the most
accurate statement.
a) Which of the following first order equations is NOT separable?
dy x 2 + y 2 x 2
=
1)
sin y + y
dx
dy x 2 y 4 + sin x
=
HONOUR HOMEWORK:
Section 4.1: 9, 21, 23, 30 [If you feel you need more practice, # 3,
6, 12, 15, 18, 24, 25, 27, 29, 33 are all doable]
Section 4.2: 7, 9, 11 [If you feel you need more practice, # 1, 2
(answer is e-x), 3, 6 (answer is e-5x), 10 (answer is
HONOUR HOMEWORK:
Section 4.3: 12, 15, 18, 27, 30, 45 [If you feel you need more
practice, # 1, 3, 5, 6, 7, 9, 11, 13, 23, 24, 25, 29, 31, 33, 35, 37, 39, 48
are all doable]
NOTE: You are responsible for section 4.4 as well in short answer,
multiple-choice
HONOUR HOMEWORK:
NOTE: On a test, I would use SI units for application questions, but
in the homework for Chapter 5, you may encounter several different
types of units. Some things to know:
Some questions use pounds and feet
If the weight of a mass is giv
HONOUR HOMEWORK:
Section 5.1 cont.: 33, 48 [Hint: For #33, use xp = e-2tcos4t 2e-2tsin4t.] [If you feel you need more practice, # 36a is also doable]
Section 5.2: 3a, 12, 27 [Note: For #27, dont worry about the first
lines of the solution in the solutions
HONOUR HOMEWORK:
Section 6.3: 6, 12, 18 [If you feel you need more practice, # 1, 3, 5,
9, 13, 15, 17, 19 are also doable.]
Good news: If you have the 6th edition of the text, all of the red and
black homework problems are the same this week. J
Final Answ
HONOUR HOMEWORK:
Section 7.1: 9, 12, 24, 30, 39 [If you feel you need more practice, #
3, 6, 15, 21, 27, 33, 36 are all doable. Please note that questions like
#24 and 30 are really important for the upcoming work were going
to do in this chapter!]
Sectio
HONOUR HOMEWORK:
These sections require PRACTICE. So, practice, practice,
practice! So, to clarify, not doing the honour homework for these
sections (especially 7.3) is a bad, very bad, VERY VERY BAD idea!
J
Section 7.3: 6, 15, 24, 48, 66 [If you feel you
HONOUR HOMEWORK:
Section 7.5: 6, 9 [If you feel you need more practice, # 3, 12
are also doable]
Section 8.1: 3, 9, 12, 18 [If you feel you need more practice, #
1, 5, 6, 7, 11, 13, 15, 17, 19 are also doable]
Good news: If you have the 6th edition of the
HONOUR HOMEWORK:
Section 8.2: 3, 9, 21, 30 [If you feel you need more practice, #
1, 5, 6, 12, 13, 19, 24, 25, 29 are also doable]
Good news: If you have the 6th edition of the text, all of the red
and black homework problems are the same this week. J
Fin
Selected Differential Equations Concepts
Eulers Method: y n 1 y n hf ( xn , y n ) , n 0, 1, 2,
Integrating Factor for Linear 1st Order:
Exact 1st Order:
Wronskian:
( x) e P ( x ) dx
M ( x, y )dx N ( x, y )dy 0
f1
f1
W [ f1 , f 2 , f n ]
f
f2
f 2
( n 1)
Laplace Transforms: HELP!
While Im sure the title is self-explanatory, basically, if youre studying Chapter 7 and
still going huh?!?, then read onthis is the worksheet for youIve organized things
by topic, so read whats useful to you, and enjoy!
Why are
MATH2860 Ilona Kletskin
1
Are you ready for MATH2860?: The Calc 2 and
Lin Alg Review Mock Test
Motivation: MATH2860 is really cool because it brings together a lot of the work
youve done in Calc II and Lin Alg. Of course, this means its super-important to
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
Solving Higher Order Ordinary Differential Equations
We will review solving higher order linear with constant coefficients, reduction of order and
variation of parameters.
Linea
HONOUR HOMEWORK:
Section 2.4: 3, 9, 24 [If you feel you need more practice, # 6, 12, 15,
18, 21, 27, 44 are all doable]
Section 3.1: 6, 9, 15, 33 [If you feel you need more practice, # 3, 12
(half life is 5600 years), 18, 21, 24, 26, 27, 36, 42, 45 are al
MATH2860U Midterm
June 5, 2007
2
1. (1 mark each; total 6 marks) In the questions below, CLEARLY circle the most
accurate statement.
a) Which of the following statements is TRUE regarding the equation:
xy 5(sin x) y = 3 y ?
1)
2)
3)
4)
The equation is non
MATH2860U Midterm
June, 2008
2
1. (2 marks each; total 8 marks) Indicate whether each of the following statements is
true (T) or false (F) and JUSTIFY your answer. The justification must be clear and
complete. You will receive credit ONLY if your answer i
Summary:FindingaGeneralSolutiontoaDifferentialEquation
Whats this?: When youre doing your homework, you know exactly which section a
given question is from, and hence, which concept youre supposed to use to solve it. So,
what do you do when you get to a t
MATH2860U: Chapter 8 cont
1
SYSTEMS OF LINEAR FIRST ORDER
DIFFERENTIAL EQUATIONS cont
Homogeneous Linear Systems (Section 8.2, pg. 311) cont.
Recall: Last day we dealt with linear systems in the case that the eigenvalues are real
and distinct.
Case 2: Rep
MATH2860U: Chapter 12
1
BOUNDARY-VALUE PROBLEMS IN
RECTANGULAR COORDINATES
Separable Partial Differential Equations (Section 12.1, pg. 433)
Recall: While we have spent the course studying ordinary differential equations, we did
also introduce partial diff
MATH2860U: Chapter 8 cont
SYSTEMS OF LINEAR FIRST ORDER
DIFFERENTIAL EQUATIONS cont
Homogeneous Linear Systems (Section 8.2, pg. 311) cont
Recall: Last day, we already learned how to solve some linear systems.
Case 3: Complex Eigenvalues
Question: What ha
MATH1020U: Chapter 9
1
DIFFERENTIAL EQUATIONS
Modelling with Differential Equations (9.1, pg. 567) cont
Recall: Last day, we introduced the idea of a differential equation.
Recall some more: We had finished off last day talking about the idea of solving a
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Int. of Rational Functions by Partial Fractions (7.4, pg. 473 )
3
Recall: We know how to integrate, for example,
But what about
4x + 5
x 2 + x 2 dx
1
x 1 + x + 2 dx
?
Question: Is there any
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
Eigenvalues and Eigenvectors
If A is an n n matrix, then a nonzero vector x is an eigenvector for A if
Ax x
for some scalar . The scalar called an eigenvalue of A.
In other word
Eigenvaluesand
Eigenvectors:Aquick
refresher
MATH2860
Row Operations
Multiply a row by a constant
Switch two rows
Add a linear combination of rows
The Process
An Example
Example cont
Example (still) cont
Laplace Transforms
used for solving linear differential equations by converting integral and differential equations into
algebraic equations which are generally easier to solve.
Laplace transforms change f (t ) function into a new function F (s ) using
Lc