Roots of Polynomials
Here are some tricks for nding roots of polynomials that work well on exams and homework assignments, where polynomials tend to have integer coecients and lots of integer, or at least rational roots.
Trick # 1
If r or r is an integer
Mathematics 256Homework #4 solutions
Exercise 1. Find the general solution to
y00 + 3y0 + y = e
The characteristic polynomial is
with roots
t + 2 sin t + 1 :
2 + 3 + 1
1 ; 2 =
=
3 2
p
=
5 2=
:
;
:
2 618
0 382
:
The solution to the full equation is
ae
Exe
A course in differential equations
Chapter 3. Second order differential equations
A second order differential equation is of the form
y00 = F (t; y; y0 )
where y = y (t). We shall often think of t as parametrizing time, y position. In this case the differ
Lecture 31:
A function f (x) on 0 < x < L can be extended to a periodic
function of period 2L by extending it to a function on L x L
which is either odd or even or neither.
If extended to be even, then its Fourier series will be a cosine series.
If extend
Lecture 34:
Teaching evaluations: open until Monday, Dec. 7.
HW10 posted, but not to be turned in. Will discuss on Friday.
TA comments on HW papers is posted on the website.
Recall: Heat equation:
2
u u
=
.
x2
t
2
In different notation:
2uxx = ut.
Recall
Lecture 26:
Two good questions were raised last time:
x0 = Ax.
1. Obtaining fundamental set of real solutions from a fundamental
set of complex solutions.
If the eigenvalues are complex, then they are of the form r = i,
and the corresponding eigenvectors
Lecture 6:
Recall defn of first order autonomous ODE: y 0 = f (y).
Recall: the direction field is constant in the t direction.
A critical point of an automonous first order ODE is a value of y
s.t. f (y) = 0.
Recall: If y0 is a critical point, y y0 is a s
Lecture 12:
Midterm on Wed.
Office Hours.
Chapter 1, Section 2.1-2.5 and 3.1-3.4.
First priority: understand assigned HW problems.
Practice with suggested problems on Nagatas website.
Old midterm?
2nd order linear ODEs:
y 00 + p(t)y 0 + q(t)y = g(t)
Metho
Lecture 20:
Example of Laplace transform method to solve ODE with discontinuous forcing term:
y 00 + y = g(t), y(0) = 0, y 0(0) = 0
where
0
0
t
<
1
g(t) = 2 1 t < 3
0 t3
Write
g(t) = 2u1(t) 2u3(t)
Recall L(uc(t) =
ecs
s .
Apply Laplace transform to ODE. u
Lecture 17:
Recall the following:
Definition: The Laplace transform of a function f (t) is:
Z
L(f ) =
estf (t)dt
0
It may be defined only for some s > a, some a.
Examples:
1
L(1) = , s > 0
s
1
L(eat) =
,s > a
sa
Let
f (t) =
Then
1 0t1
0 t>1
1 es
, s>0
L(
LECTURE 1:
ADMIN:
Math 256: Differential Equations
Most all info is on the web page:
http:/www.math.ubc.ca/marcus/Math256
Primarily an introduction to ordinary differential equations
One chapter on partial differential equations
Course material will mo
Lecture 3:
Last time: found general solution of any first order linear ODE:
y 0 + p(t)y = g(t) (standard form)
R
y(t) = (1/(t)( (t)g(t)dt + C)
where (t) = e
R
p(t)dt
(the integrating factor).
Most important thing to remember is the equation for the integr
Lecture 9:
Continue a digression into exponentials of complex numbers.
Recall: for real t:
t
e =
n
X
t
n=0
n!
For real t, we define
it
e :=
X
(it)n
n=0
n!
(it)2 (it)3 (it)4
= 1 + it +
+
+
+ .
2
3!
4!
t2 it3 t4
= 1 + it
+ + .
2
3! 4!
Separate the real an
Lecture 23:
Recall that we are considering homogeneous linear systems with
constant coefficients:
x0 = Ax
The entries of the matrix A are constant and the entries of vector x
are functions of t.
Will usually assume det (A) 6= 0.
Recall Example 1:
2
0
x0 =
Mathematics 256Homework #5 solutions
Exercise 1. A system satisfies a differential equation of the form
y00 + cy0 + y = 0
(a) Graph the quasi-period as a function of c. (b) For what value of c is it equal to twice that of the free period?
(c) For what val
A course in differential equations
Chapter 2. Numerical methods of solution
We shall discuss the simple numerical method suggested in the last chapter, and explain several more sophisticated
ones. We shall also deal with the problem of estimating error in
Mathematics 256Homework #3 solutions
20 steps from t = 0 gave an estimate for y(1) of
0:117820, and a run with 40 steps gave an estimate of 0:118142 What is the approximate error in the second
run, and what step size should you use to get an accuracy of 6
Simple ODE Solvers - Error Behaviour
These notes provide an introduction to the error behaviour of Eulers Method, the
improved Eulers method and the RungeKutta algorithm for generating approximate solutions to the initial value problem
y (t) = f t, y (t)
Derivation of the Wave Equation
In these notes we apply Newtons law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the wave equation. Consider a tiny element of the string. T (x + x, t) (x + x, t) u (x, t) x
Richardson Extrapolation
There are many approximation procedures in which one rst picks a step size h and
then generates an approximation A(h) to some desired quantity A. Often the order of the
error generated by the procedure is known. In other words
A =
Properties of Exponentials
In the following, x and y are arbitrary real numbers, a and b are arbitrary constants that are
strictly bigger than zero and e is 2.7182818284, to ten decimal places.
1) e0 = 1, a0 = 1
2) ex+y = ex ey , ax+y = ax ay
3) ex =
4) e
A Lightning Fast Review of Determinants
Let A be an n n matrix. Then, det A is the number given by
n
Aij (1)i+j det Mij
det A =
j =1
n
Aij (1)i+j det Mij
=
i=1
In the upper formula 1 i n is any xed row. The upper formula is called expansion
along the ith
A Lightning Fast Review of Eigenvalues and Eigenvectors
Let A be an nn matrix. Then, by denition, v is an eigenvector of A with eigenvalue
if and only if
(i) v = 0
(ii) Av = v
Fix for a moment. Then (A I )v = 0 has a nonzero solution v if and only if the
How Systems of First Order ODEs Arise
A system of rst order ordinary dierential equations is a family of n ordinary
dierential equations in n unknown functions y1 (t), , yn (t). To be ordinary (as opposed
to partial) dierential equations, only ordinary (a
Linear Regression
Imagine an experiment in which you measure one quantity, call it y , as a function of a
second quantity, say x. For example, y could be the current that ows through a resistor when
a voltage x is applied to it. Suppose that you measure n
Complex Numbers and Exponentials
A complex number is nothing more than a point in the xy plane. The sum and
product of two complex numbers (x1 , y1 ) and (x2 , y2 ) is dened by
(x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 )
(x1 , y1 ) (x2 , y2 ) = (x1 x2
A course in differential equations
Chapter 1. First order differential equations
We begin the course by examining a reasonably realistic physical example.
1. Flow in and out of a tank
Consider a cylindrical tank of water with water flowing into it at the
A course in differential equations
Chapter 6. Partial differential equations
Physical systems one really wants to understand are often essentially continuous in nature, resembling systems
with an infinite number of componentsfor example, an electric field
Mathematics 256Homework #1 solutions
Exercise 1. Assume water flows into a cylindrical tank of cross section A with a flow of f litres per second, and
flows out at the bottom through a hole of effective area .
(a) Write down the differential equation desc