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
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)
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 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 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 )
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
MAT244 Fall 2016 Term Test 1
Problem 1 Find the solution of the following problem
y 0 + 4y = xex
.
y(0) = 1
Problem 2 Solve the following problem
y(x + y) + (xy + 1)y 0 = 0
,
y(0) = e
given that the equation has an integrating factor of the form = (y).
Pr
Problem 1
This is a linear 1st order ODE, so we seek Ran integrating
factor. For a linear equation y 0 + q(x)y =
g(x), the integrating factor is (x) = exp q(t)dt , so we compute:
Z
4dt = e4x .
(x) = exp
Multiplying both sides of the equation by , we get:
Term Test, February 25, 2016
Problem 1: ANS: a)
MAT 244 - Ordinary Differential Equations
y 0 + p(x) y = q(x)y n .
b). The equation can be written as
y0
1
y = (1 + ln x) y 3 .
x
The substitution v = y 2 reduces the problem to
2
dv
+ v = 2(1 + ln x).
dx
Easy Problems
Problem 1. If y1 and y2 are two solutions of the equation
y + cos (x) y + ex y = 0,
show that
W (y1, y2)(0) = W (y1, y2)().
Problem 2. Assume that a1, a2 are continuous functions. If y1, y2 are two linearly independent solutions of the equat
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 ( ()
University of Toronto, 2016, MAT244, Ordinary Differential Equations
Problem 5
Put X = x + 1 and Y = y + 1. Then
dy
dY
dY
dY dX
dY
dY
=
=
=
.
dx
dx
dy
dX dx
dy
dX
Substituting this into the given ODE, we get
Y X
dY
Y /X 1
dY
=
=
dX
Y +X
dX
Y /X + 1
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1
1 Linear homogeneous systems
1 Linear homogeneous systems
1.1 Two real distinct eigenvectors
If (1, Sv1) and (2, Sv2) are eigenpairs of A then the general homogeneous solution to the
system Sy = Ay
S is
Syh(x) = c1 e x Sv1 + c2 e x Sv2.
1
2
Trajectories
MAT244H1F1: Introduction to Ordinary
Differential Equations
First Midterm, Question 4 Solution
Problem 1 Consider the equation y 0 = y y 3 .
1. Find all equilibria of the equation.
2. Draw the phase line and determine the stability of each equilibrium.
3.
Lecture 9
IMC 200 Innovation and Entrepreneurship
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