MTH U345
Ordinary Dierential Equations
Fall 2008
Lab 1: Partial solution
Exercise 2(d) Consider the initial value problem
dy/dt = y (y 2),
y (0) = 3.
What happens to y (t) as t increases? Can you show that y (t) escapes to innity?
First, solve the ODE by
MTH U345
Ordinary Dierential Equations
Fall 2008
Practice Quiz 2
1. Convert the following second order D.E. to a system of rst order D.E.s. DO NOT TRY TO
SOLVE the system. All you have to do is write out the rst order system.
y (t) = 7y (t) + 2y (t) + 3y
MTH U345
Ordinary Dierential Equations
Fall 2008
Lab 1: Slope Fields and Solution Curves
The equation y = f (x, y ) determines a slope eld (or direction eld) in the xy -plane:
the value f (x, y ) is the slope of a tiny line segment at the point (x, y ). I
MTH U345
Ordinary Dierential Equations
Fall 2008
Quiz 1
1. 10 points Solve the initial value problem
dy
= t + ty 2 ,
dt
For which values of t is the solution dened?
2. 10 points Given the dierential equation
y (0) = 1
dy
= (y 2 4)(y 3).
dt
(a) Sketch the
MTH U345
Ordinary Dierential Equations
Practice Quiz 5
1. Find the Laplace transforms F (s) of the following functions f (t):
(a) f (t) =
0,
(t 2)2 ,
t<2
t2
(b) f (t) =
0,
t<1
2 2t + 2, t 1
t
(c) f (t) = u2 (t)e3t6
(d) f (t) = t u1 (t)(t 1)
(e) f (t) = (t
Name:
MTH U345
Ordinary Dierential Equations
Fall 2008
Quiz 2
1. 5 points Convert the following second order dierential equation to a system of rst order
dierential equations. DO NOT TRY TO SOLVE the system.
y (t) = 2y (t) 5y (t) + 7y 3 (t).
2. 5 points W
Name:
MTH U345
Ordinary Dierential Equations
Quiz 3
1. 10 points Consider the linear system Y = AY , with A =
5 2
.
12
(a) Find the eigenvalues 1 and 2 of the matrix A.
(b) Find (non-zero) eigenvectors V1 and V2 corresponding to each eigenvalue.
(c) Find
Name:
MTH U345
Ordinary Dierential Equations
Fall 2008
Quiz 4
1. 9 points Find the general solution of the dierential equation y 4y 5y = 6e2t .
2. 9 points Consider the dierential equation y + 16y = cos(4.1t).
(a) Determine the frequency of the beats.
(b)
MTH U345
Ordinary Dierential Equations
Fall 2008
Lab 3: Using MATLAB for Dierential Equations 1
We are now familiar with using a spreadsheet to set up numerical methods for approximating solutions of a dierential equation. In this computer lab, we shall n
MTH U345
Ordinary Dierential Equations
Fall 2008
Lab 2: Numerical Methods of Euler
A numerical method for approximating the solution of the initial-value problem
(*)
y
= f (x, y )
y ( a) = y 0
involves replacing the continuous variable x by a set of discr
MTH U345
Ordinary Dierential Equations
Fall 2008
Practice Quiz 3
2 3
.
1 6
(a) Find the eigenvalues 1 and 2 of the matrix A.
1. Consider the linear system Y = AY , with A =
(b) Find (non-zero) eigenvectors V1 and V2 corresponding to each eigenvalue.
(c) F
MTH U345
Ordinary Dierential Equations
Fall 2008
Practice Quiz 1: Answer Key
1.
2.
3.
4.
5.
6.
dy 2
y = t2 sin t y (t) = t2 (k cos t), for t = 0
dt
t
dy
(t + 1)
= 1 + y 2 y (t) = tan(ln |1 + t| + k ), for t = 1
dt
1
1
t
k
dx
+ x=
y (t) = + , for t > 0
d
MTH U345
Ordinary Dierential Equations
Fall 2008
Quiz 1
dy
= t + ty 2 , y (0) = 1. For which values of t is the solution dened?
dt
dy
dy
t2
t2
= t(1 + y 2 )
= tdt + C arctan y =
+ C y = tan
+C
dt
1 + y2
2
2
y (0) = 1 tan(C ) = 1 C =
4
t2
The solution cu
MTH U345
Ordinary Dierential Equations
Fall 2008
Practice Quiz 2: Answer Key
1. Solve the initial value problem 4y 12y + 9y = 0, y (0) = 9, y (0) = 8.
y (t) = 9e3t/2
11 3t/2
te .
2
x1
6 4
is a solution of X = AX where A =
, give x1 and x2 at the point wh
MTH U345
Ordinary Dierential Equations
Fall 2008
Solutions to Practice Quiz 5
2. Find the inverse Laplace transform f (t) of the following functions F (s):
1 2s
(c) F (s) = 2
f (t) = e2t (2 cos(2t) + 5 sin(2t)
s + 4s + 5
5
2s 3
f (t) = et (2 cos(3t) sin
MTH U345
Ordinary Dierential Equations
Fall 2008
Solutions to Quiz 5
1. Find the Laplace transforms F (s) of the following functions f (t):
(a) f (t) =
0,
t<3
2 6t + 1 , t 3
t
f (t) = u3 (t)(t 3)2 8)
F (s) = e3s
2
8
3
s
s
(b) f (t) = e4t 3 (t) e2t2 u1 (t)
1
2
3
4
5
6
7
8
9
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH U345
FINAL EXAM
Fall 2008
Put your name in the blanks above. Put your nal answers to each question in the designated spaces.
Calculators are permitted. A single sheet of formula
MTH U345
Ordinary Dierential Equations
Fall 2008
Practice Quiz 4
1. Find the general solution of the dierential equation y 7y + 12y = 5e3t .
2. Solve the initial value problem y 3y + 2y = t, y (0) = 0, y (0) = 0.
3. Solve the initial value problem y + 4y
Name:
MTH U345
Ordinary Dierential Equations
Quiz 5
1. 12 points Find the Laplace transforms F (s) of the following functions f (t):
(a) f (t) =
0,
t<3
2 6t + 1 , t 3
t
(b) f (t) = e4t 3 (t) e2t2 u1 (t)
2. 12 points Find the inverse Laplace transform f (t