LECTURE 12: MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 05, 2015
RLC Circuit System
There is a resistor with a resistance of R ohms, an inductor with an inductance of L henries, and a capacitor with
capacitance of C farads. There is also an electric sourc
LECTURE 4: FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS AND THE
INTEGRATING FACTORS
MINGFENG ZHAO
September 16, 2015
Theorem of the existence and uniqueness for the rst order dierential equations with
the initial value condition
Theorem 1 (Picards Theorem on
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS, AND AUTONOMOUS
EQUATIONS
MINGFENG ZHAO
September 23, 2015
To solve an exact equation
Let M (x, y) + N (x, y)y = 0 be an exact equation, that is, My (x, y) = Nx (x, y), then we can nd (x, y) such that
x (x
LECTURE 6: AUTONOMOUS EQUATIONS AND EULERS METHOD
MINGFENG ZHAO
September 25, 2015
Phase diagram of the autonomous equation
An autonomous equation is of the form y = f (y), if f (a) = 0, y(x) a is called an equilibrium solution, and a is
called a critical
LECTURE 10: HOMOGENEOUS SECOND ORDER LINEAR ODES WITH CONSTANT
COEFFICIENTS
MINGFENG ZHAO
September 30, 2015
Theorem 1. Let p(x) and q(x) be continuous functions, y1 and y2 are two linearly independent solutions to a homogeneous equation y + p(x)y + q(x)y
Math 215/255 Final Exam (Dec 2005)
Last Name:
First name:
Student #:
Signature:
Circle your section #:
Burggraf=101, Peterson=102, Khadra=103, Burghelea=104, Li=105
I have read and understood the instructions below:
Please sign:
Instructions:
1. No notes
MIDTERM EXAM TOPICS: MATH 215/255 FEBRUARY 2016
Note: Any topic from any part of the course could be tested either individually or in combination with
other course topics.
First Order Differential Equations
Identify linear, nonlinear and separable equati
HOMEWORK 5: MATH 215 Winter 2016
1. The functions sinh(t) and cosh(t) are defined by
sinh(t) =
1 t
e et ,
2
cosh(t) =
1 t
e + et .
2
You should check that you agree that sinh(0) = 0, cosh(0) = 1, cosh(t) = cosh(t), sinh(t) = sinh(t)
and
d
d
sinh(t) = cosh
Math 257/316, Midterm 1, Section 102
4 pm on 8 th October 2014
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
A formula sheet is provided.
Maximum score 50.
1. Consider the second order dierential
Be sure this exam has 5 pages including the cover
The University of British Columbia
MATH 215/255, Section 104
Midterm Exam I October 2014
Signature
Name
Student Number
This exam consists of 4 questions worth 10 marks each. No notes nor calculators.
Probl
LECTURE 2: SEPARABLE DIFFERENTIAL EQUATIONS
MINGFENG ZHAO
September 11, 2015
Example 1. Verify that x(t) = Ce2t is a solution to x = 2x. Find C to solve for the initial condition x(0) = 100.
In fact,
x = 2Ce2t = x = 2x = x(t) = Ce2t is a solution to x = 2
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS
MINGFENG ZHAO
September 21, 2015
Exact equations
Denition 1. Consider the dierential equation M (x, y)+N (x, y)y = 0, we say that the dierential equation M (x, y)+
N (x, y)y = 0 is exact if My (x, y) = Nx (
LECTURE 1: INTRODUCTION TO THE DIFFERENTIAL EQUATIONS
MINGFENG ZHAO
September 09, 2015
Introduction to the dierential equations
Denition 1. A dierential equation is an equation involving the derivatives of the dependent variable.
If the
highest order of t
LECTURE 17: FORCED OSCILLATIONS AND RESONANCE
MINGFENG ZHAO
October 21, 2015
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
We are interested in periodic forcing, F (t) = F0 cos(t) wi
LECTURE 16: NONHOMOGENEOUS EQUATIONS AND FORCED OSCILLATIONS AND
RESONANCE
MINGFENG ZHAO
October 19, 2015
Theorem 1. Let yc (x) be the general solution to the homogeneous equation y + p(x)y + q(x)y = 0 ( which is call the
complementary solution), and yp (
LECTURE 23:THE LAPLACE TRANSFORM AND SYSTEMS OF ODES
MINGFENG ZHAO
November 02, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
L[f (t)](s) =
f (t)est dt, fo
LECTURE 19: THE LAPLACE TRANSFORM AND TRANSFORMS OF DERIVATIVES AND
ODES
MINGFENG ZHAO
October 26, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est dt,
LECTURE 20: THE LAPLACE TRANSFORMS OF INTEGRAL, CONVOLUTION AND DIRAC
DELTA
MINGFENG ZHAO
October 28, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est
LECTURE 18: THE LAPLACE TRANSFORM AND TRANSFORMS OF DERIVATIVES AND
ODES
MINGFENG ZHAO
October 23, 2015
Example 1. A mass of 4 kg on a spring with k = 4 and a damping constant c = 1. Suppose F0 = 2. Using forcing
function F0 cos(t), nd the that causes pra
LECTURE 21: THE LAPLACE TRANSFORMS OF INTEGRAL, CONVOLUTION AND DIRAC
DELTA
MINGFENG ZHAO
October 30, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est
LECTURE 13: MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 07, 2015
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
For free motion (that is, F (t) = 0), rewrite the equation, we have
2
x
Be sure this exam has 5 pages including the cover
The University of British Columbia
MATH 215/255
Midterm Exam I October 2015
Name
Signature
Student Number
Course Number
Circle Section:
101 Zhao
102 Tsai
103 Kolokolnikov
104 Zhao
This exam consists of 4 q
Be sure this exam has 5 pages including the cover
The University of British Columbia
MATH 215/255
Midterm Exam I October 2015
Name
Signature
Student Number
Course Number
Circle Section:
101 Zhao
102 Tsai
103 Kolokolnikov
104 Zhao
This exam consists of 4 q
Math 257/316 Final Exam, April 21 2015
Last Name:
First Name:
Student Number:
Signature:
Instructions. The exam lasts 2.5 hours. No calculators or electronic devices of any kind are
permitted. A formula sheet is attached. There are 12 pages in this test i
Dec. I7, 2012 Math 257/316 Name: Page 2 of 17 pages
I. Consider the differential equation
I 14; 8x2y”+2my'+(l+2m)y=0 (1)
(a) Classify the points 0 S a: < 00 as ordinary points, regular singular points, or irregular singular points.
N
(b) Find two values o
Announcements:
Quiz on Wednesday March 23, topics: nonlinear
systems, 2nd order ODEs
WW8 posted, due March 25, 9 am
Office hours this week: Tuesday 2 - 4 pm
This weeks plan: Numerical methods
Tuesday, 22 March, 16
Slope (direction) field - Can we use
Math 215/255: Homework #4 Due Friday, March 17, 2017.
Turn in problems in correct order and STAPLE your work!
1. Exponential matrix.
Compute etA for the following matrices:
(a)
1
1
1
1
1
3
0
1
A1 =
(b)
A2 =
Solution: Problem 1:
(a) In order to exponentiat
HOMEWORK 3: Math 215/255 February 2017
Part1: practice problems for the midterm, do not turn in this part.
Part2: Due in class Monday February 27
Part1
1. Find the solution y(t) to
0
y =
3 2
1 2
y,
with y(0) =
3
3
.
Also, describe the behavior of the solu
HOMEWORK 6: MATH 215
Problem 1: Determine the Laplace transform F (s) =
gral:
R
0
est f (t)dt of the following by evaluating the inte-
1. f1 (t) = te3t
2. f2 (t) = et sin(2t) + cos(3t)
Solution:
1. Y1 (s) =
1
(s3)2
2. Y2 (s) =
s
s2 +9
+
2
(s+1)2 +4
Proble
- University of British Columbia
Math 215/255
Midterm 1 Date: October 3, 2016
Name (print):
Student ID Number:
Signature: SOL uT/U/VS /tD/A/cfw_ SCHEV/IE
Instructions:
1. No notes, books or calculators are allowed.
2. Read the questions carefully and make
Math 215/255 Final _ ' Name:
Problem IV.
Consider the function
shown in the gure below.
00041
[15 marks]
Page 8 of 14 Meth 215/255 Final Name:
" 00041
(b ) (10 marks) Using Laplace transforms, solve the following initial value problem:
0 + 40 E W)