Second Order Linear Differential Equations
Second order linear equations with constant coefficients; Fundamental
solutions; Wronskian; Existence and Uniqueness of solutions; the
characteristic equation; solutions of homogeneous linear equations;
reduction
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The University of British Columbia
Sessional Examinations - April 2008
Mathematics 256
Dierential Equations
Time: 2 1 hours
2
Closed book examination
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April 2009
MATH 256
Name
Page 2 of 11 pages
Marks
[20] 1.
Linear dierential equations.
(a) Solve the initial value problem:
y + 3y = 2e2t ,
y(0) = 1.
(b) Solve the initial value problem:
2
y + 2ty = et ,
y(0) = 3.
Continued on page 3
April 2009
MATH 256
N
Math 256 Section 201 Final Exam
Spring 2007
Instructor: Paul A.C. Chang
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Be sure that this examination has 11 pages including this cover
The University of British Columbia
Sessional Examinations - April 2010
Mathematics 256
Dierential Equations
Time: 2 1 hours
2
Closed book examination
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Signature
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Instructors
The University of British Columbia
Final Examination April 24, 2013
Math 2 56
Closed Book Examination Time: 20 minutes
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MATH 256 Midterm 1 February 10, 2015.
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differ slightly in the final marking scheme.
1. [5 pts] The solution to the equat
Modeling with First Order Equations
1. Construction of the Model. Translate the physical situation into mathematical terms. Pay attention to the units of the quantities, and use one unit
for one type of quantities.
2. Analysis of the Model. Solve the dier
List of Formulas and Equations
1. First order equations
1.1) The solutions to
y + p(t)y = 0,
y = Ce
is
p(t)dt
1.2) The steps to solve
y + p(t)y = g(t)
are
compute (t) = e
y(t) =
p(t)dt
1
(C +
(t)
,
(t)g(t)dt =
(t)g(t)dt)
1.3) Separable equation
dy
= h(t)k
Math 146C Homework #2
Solutions To Graded Problems
10.7:
10.5:
10.6:
10.8:
6,9,13,14,21
3,6,8,9,13,14,20,22
4,5,10,12,21
1,3
Section 10.7
Problem 6 - Consider an elastic string of length L whose ends are held
xed. The string is set in motion from its equi
HW 9 due Nov 2, Fri
Dene: Denote the translation operator Ta , i.e.,
Ta f (t) = f (t a)
1. Solve initial value problems.
(a) y + y = f (t),
y (0) = 0, y (0) = 1;
f (t) =
f (t) = u(t) u3 (t)
Lcfw_y + y = Lcfw_u(t) u3 (t)
(s2 + 1)Y = 1/s e3s /s + 1
1
1
Y =
f ( t ) = L -1 cfw_F ( s )
1.
1
s
n!
s n +1
1
3.
Table of Laplace Transforms
F ( s ) = L cfw_ f ( t )
t n , n = 1, 2,3,K
5.
sin ( at )
9.
t sin ( at )
sin ( at ) - at cos ( at )
(s
2
+ a2 )
22
2
2
13.
cos ( at ) - at sin ( at )
15.
sin ( at + b )
17.
sinh
Math 54 Spring 2005
Solutions to Homework Section 3.2
April 8th, 2005
1. Find the Wronskian of the functions e2t and e3t/2 .
Det
e2t e3t/2
2e2t 3 e3t/2
2
3
=e2t ( 2 e3t/2 ) e3t/2 (2e2t ) = 7 et/2 .
2
2. Find W (cos t, sin t)(t).
cos t sin t
sin t cos t
=
Math 54, Spring 2005
10.5 and 10.6 Homework Solutions
May 9th
Section 10.5
In Problems 1 and 3, determine whether the method of separation of variables can be used to replace the
given partial dierential equation by a pair of ordinary dierential equations
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The University of British Columbia
MATH 256, Sections 102 and 103
Final Exam December 4, 2014
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Math 256 Final examination
University of British Columbia
April 21, 2016, 3:30 pm to 6:00 pm
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Math 256 Final examination
University of British Columbia
April 25, 2014, 12:00 pm to 2:30 pm
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