1
Eigenvector Reduction Method
MAT244: Introduction to ODEs
summary of the last topic
1 Eigenvector Reduction Method
The general solution to the system
Sy = Ay
S
(1)
Case 1. If 1 2 then use formula
Sy (x) = e1xa1Sv1 + e2xa2Sv2,
(2)
where Sv1, Sv2 are eige
MAT244F ODEs
Midterm #1,
Oct. 12, 2011
6:10-7:00pm
Total: 20 points. No aids allowed!
1. (4 pts) Solve the following initial value problem.
y + y = ty 3
y(0) = 1.
(1)
2. (a) (4 pts) Solve the following initial value problem.
(t2 1)y + 2y = t3 + 2t2 + t,
y
MAT244, 2014F, Solutions to Term Test 2
Problem 1. Solve the following initial value problem
x3 y 3x2 y + 6xy 6y = 24x1 + 288 ln x ,
y(1) = 0,
y (1) = 7,
y (1) = 22 .
Solution. It is Eulers equaltion with the characteristic polynomial
r(r 1)(r 2) 3r(r 1)
MAT 244-Term Test 1 Solution
Problem 1
Find the solution to the following problem
" + 4 = (
0 =1
Solution:
This is a linear 1st order ODE, so we seek an integrating factor. For a linear equation y + q(x)y
= g(x), the integrating factor is (x) = exp ( ()
MAT244, Spring 2013, Term Test 1, Wednesday, February 13, Solutions 1
Please note: handouts are not only obligatory but allow you to solves problems faster. See problem 3 in particular.
1 [6 points]. Find integrating factor and solve
x dx + y(1 + x2 + y 2
MAT 244 Ordinary Dierential Equations, Term Test #1, solutions
1
MAT 244 Ordinary Dierential Equations
Term Test # 1, February 3, 2010, 8:10-8:50 pm
1 (8 pts) Find the general solutions of the dierential and solve the initial value
problem
x + y sin(xy )
MAT244, 2014F, Solutions to MidTerm
Problem 1. If exists, nd the integrating factor (x, y) depending only on
x , only on y and on x y justifying your answers and then solve the ODE
3x +
6
y
+
x2 3y
+
y
x
y =0.
Also, nd the solution satisfying y(1) = 2 .
S
MAT 244, Fall 2013. Midterm test,
SOLUTIONS.
1. (20 pts)
a) Write a dierential equation describing a function y(x) with the
following property: the slope of the tangent to the graph at a point
(x, y) is the product of the coordinate x and the square of th
Department of Mathematics
Page 1 of 17
Final exam
Term: Summer 2013
Student ID Information
Last name:
First name:
Student ID #:
Course Code:
Mat 244
Course Title:
Introduction to Ordinary Dierential Equations
Instructor:
Jordan Bell
Date of Test:
Time Per
2nd Order Linear ODEs
Theory of 2nd Order Linear Homogeneous ODEs
MAT244 Lecture 7: Second Order ODEs
Tyler Wilson
June 7, 2017
Tyler Wilson
MAT244 Lecture 7: Second Order ODEs
2nd Order Linear ODEs
Theory of 2nd Order Linear Homogeneous ODEs
2nd Order Li
First Order Linear Differential Equations
MAT244 Lecture 2: Linear Equations
Tyler Wilson
May 19, 2017
Tyler Wilson
MAT244 Lecture 2: Linear Equations
First Order Linear Differential Equations
First Order Linear Differential Equations
Tyler Wilson
MAT244
1. Using y = emx a a trial solution, derive the characteristic equation for
a2 y 00 + a1 y 0 + a0 y = 0.
2. Find the general solution to
2y 00 7y 0 + 3y = 0
3. Find the general solution to
y 00 + 2y 0 = 0
4. Show Ly = y 00 + p(x)y 0 + q(x)y is a linear op
Separable Differential Equations
First Order Models
MAT244 Lecture 2: Separable Differential
Equations and Modeling
Tyler Wilson
May 24, 2017
Tyler Wilson
MAT244 Lecture 2: Separable Differential Equations and Modelin
Separable Differential Equations
Firs
Introduction
Direction Fields
Initial Value Problems
MAT244 Lecture 1: Introduction and Direction
Fields
Tyler Wilson
May 17, 2017
Tyler Wilson
MAT244 Lecture 1: Introduction and Direction Fields
Introduction
Direction Fields
Initial Value Problems
Introd
Eulers Method
MAT244 Lecture 6: Eulers Method
Tyler Wilson
June 2, 2017
Tyler Wilson
MAT244 Lecture 6: Eulers Method
Eulers Method
Eulers Method
Tyler Wilson
MAT244 Lecture 6: Eulers Method
Eulers Method
Eulers Method
So far we have seen a couple differen
Non-Homogeneous 2nd Order Linear Equations
Undetermined Coefficients
Variation of Parameters
MAT244 Lecture 10: Non-Homogeneous 2nd
Order Linear Equations
Tyler Wilson
June 21, 2017
Tyler Wilson
MAT244 Lecture 10: Non-Homogeneous 2nd Order Linear Equat
No
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Harmonic Motion
MAT244 Lecture 11: Harmonic Motion
Tyler Wilson
June 23, 2017
Tyler Wilson
MAT244 Lecture 11: Harmonic Motion
Harmonic Motion
Harmonic Motion
Tyler Wilson
MAT244 Lecture 11: Harmonic Motion
Harmonic Motion
Spring and Mass System
Tyler Wils
MAT244F ODES Midterm #1, Oct. 141, 2015 6:107:00pm
Total: 20 points. No aids allowed!
@4 pts) Find the solution of
m(m+1)y+y+1=0
satisfying y(1) = 1.
@onsider a Bernoulli equation
d
142% 2962/,
dx y 931
(2 pts) Using the substitution 2 = y2, reduce thi
MAT244- TermTest I
Date: October 12, 2016
Time: 50 min
Surname: _Given Name:_
Id Number:_
INSTRUCTIONS:
Answer all questions
Unsubstantiated work may not receive full credit
No aids are permitted
Question
Maximum Grade Your Grade
1
10
2
10
3
10
4
10
5
10
FACULTY OF ARTS & SCIENCES
University of Toronto
MAT244
Ordinary Differential Equations
Exam, February 25, 2016
Examiners: N. Hoell, M. Niksirat, & Y. Song
Duration: 90 minutes
NO AIDS ALLOWED.
Total: 100 marks
Family Name:
(Please Print)
Given Name(s):
(
Linear vs Non-linear First Order ODEs
MAT244 Lecture 4: Linear vs Non-Linear First
Order ODEs
Tyler Wilson
May 26, 2017
Tyler Wilson
MAT244 Lecture 4: Linear vs Non-Linear First Order ODEs
Linear vs Non-linear First Order ODEs
Linear vs Non-linear First O