5
Exact equations
In Lecture 3, where I discussed the geometric interpretation of the rst order ODE, an attentive
reader should have noted that the relation between the integral curves as the curves tangent to the
given direction eld and the graphs of the
9
Numerical methods
What to do if analytical solution is not possible, and qualitative methods, that we studied in the
last lecture, are not enough. In this case it is always possible to apply some numerical methods to
solve the problem. By numerical meth
10
Homogeneous second order linear ODE with constant coecients
This lecture opens second part of our course. From now on the main object of the study will be the
linear ODE. And even not simply linear, but linear ODE with constant coecients. In these lect
15
Laplace transform. Basic properties
We spent a lot of time learning how to solve linear nonhomogeneous ODE with constant coecients.
However, in all the examples we consider, the right hand side (function f (t) was continuous. This is
not usually so in
10
Complex numbers. Solving homogeneous second order linear
ODE with constant coecients
This lecture opens the second part of our course. From now on the main object of the study will be the
linear ODE. And even not simply linear, but linear ODE with cons
MATH266: Exam II, October 29
Name:
1. Solve the following ODE using the method of undermined coecients
y 2y = 2e2t
2. Solve the following IVP using variation of parameters:
y + y =
1
,
sin t
y(/2) = 0, y (/2) = /2
3. An object of mass 1 slug is attached t
MATH266: Exam I
Name:
1. Find a solution to the IVP
dy
y
= 2
,
dx
x +1
y(0) = 2.
2. Solve the dierential equation
x
3. Solve the equation
1
dy
+ y = xy 2 .
dx
(
)
2t3
y
6t (1 + ln y)dt e
dy = 0.
y
2
1
Math266: Exam I (page 2) Name:
4. When a cup of coee
MATH266: Final Exam, December 10 (2323)
Name:
Instructions
Show work for partial credit
Box your answer
No calculators allowed
A standard 8 1/2 by 11 sheet of paper with students notes is allowed
Problems
1. Solve the IVP
yy = t,
y(2) = 1
2. The tempe
MATH266: Exam I
Name:
1. Solve the dierential equation
t
dy
= y tey/t .
dt
2. Solve the dierential equation
dy
t
+ 2y = 2 .
dt
y
3. Solve the following IVP
3t2 y 2 dt + (1 + 2t3 y)dy = 0,
1
y(1) = 2
Math266: Exam I (page 2) Name:
4. Suppose that the half
Lecture 3 learning objectives
Define natural selection using Darwins 4
postulates
Interpret graphs that depict trait variation and
heritability
Synthesize and evaluate data to determine
whether or not evolution by natural selection is
occurring
Descri
11
Elements of the general theory of the linear ODE
In the last lecture we looked for a solution to the second order linear homogeneous ODE with constant
coecients in the form y (t) = C1 y1 (t) + C2 y2 (t), where C1 , C2 are arbitrary constants and y1 (t)
12
Solving nonhomogeneous equations: Method of educated guess
Consider nth order linear ODE with constant coecients
y (n) + an1 y (n1) + . . . + a1 y + a0 y = f (t).
(1)
We know by now that the general solution to this equation can be represented in the
4
Solving rst order linear ODE. Newtons law of cooling
Linear equations and systems will take a signicant part of the course. Here we start with the simplest
linear problem:
Denition 1. The rst order ODE of the form
y + p(x)y = q (x)
(1)
is called linear.
3
Geometric interpretation of the rst order ODE. Existence and
uniqueness theorem
It was stated that our main goal for the rst half of the course is to learn analytical methods of solving
ODE. Especially, of solving the rst order ODE in the form
y = f (x,
Introductory words
These lecture notes were prepared during Fall 2013 with the goal to have a selfcontained exposition
of the course content how I present it during the actual lecture hours. Therefore, these lecture notes
include only the material which
2
Separable equations
We now start to systematically study rst order ODE of the form
y = f (x, y ),
(1)
where f is a given function of two variables, x is an independent variable and y = y (x) is the dependent
variable or our unknown function. It is usual
6
Substitutions I
Consider again the rst order ODE in the form
M (x, y ) dx + N (x, y ) dy = 0.
(1)
We learnt that we can solve (1) if it is either separable, linear, or exact. In some cases an integrating
factor can be found. Let us reiterate the denitio
7
Substitutions II
Some of the topics in this lecture are optional and will not be tested at the exams. However, for a
curious student it should be useful to learn a few extra things about ordinary dierential equations.
7.1
Examples
If it is possible to s
8
Autonomous rst order ODE
There are dierent ways to approach dierential equations. Prior to this lecture we mostly dealt with
analytical methods, i.e., with methods that require a formula as a nal answer. Another possible
approach is the numerical method
13
Solving nonhomogeneous equations: Variation of parameters
We are still solving
Ly = f,
(1)
where L is a linear dierential operator with constant coecients and f is a given function. Together
(1) is a linear nonhomogeneous ODE with constant coecients, w
14
Mass on a spring and other systems described by linear ODE
14.1
Mass on a spring
Consider a mass hanging on a spring (see the gure). The position of the mass in uniquely dened by
one coordinate x(t) along the xaxis, whose direction is chosen to be alo
Study Guide for Exam 2:
Female fitness is due to how many eggs can produce
Male fitness is dependent on how many eggs they can fertilize
Thus explaining why sexual selection acts on males more strongly than females
Lectures 9 & 10: Quantitative genetics
