First Order Difference Equations
Differential equation are great for modeling situations where there is a continually changing
population or value. If the change happens incrementally rather than continuously then
differential equations have their shortco
MATH 204 PRACTICE EXAM I
Please work out each of the given problems on your own paper. Credit will be
based on the steps that you show towards the final answer. Show your work.
Problem 1
Solve the following differential equations.
A. (25 points)
x+y
y' =
Separable Differential Equations
Definition and Solution of a Separable Differential Equation
A differential equation is called separable if it can be written as
f(y)dy = g(x)dx
Steps To Solve a Separable Differential Equation
To solve a separable differe
Modeling with First Order Differential Equations
Whenever there is a process to be investigated, a mathematical model becomes a possibility.
Since most processes involve something changing, derivatives come into play resulting in a
differential equation.
First Order Linear Differential Equations
In this section we will concentrate on first order linear differential equations. Recall that this
means that only a first derivative appears in the differential equation and that the equation is
linear. The gener
Name
MATH 204 PRACTICE FINAL
Please work out each of the given problems on your own paper. Credit will be based on the steps
that you show towards the final answer. Show your work.
Problem 1
Solve the following differential equations.
A. (x2 + 1)y' + 4xy
A Sketch of the Proof of the Existence and Uniqueness
Theorem
Recall the theorem that says that if a first order differential satisfies continuity conditions, then
the initial value problem will have a unique solution in some neighborhood of the initial v
Autonomous Equations and Population Dynamics
Definition
A differential equation is called autonomous if it can be written as
dy/dt = f(y)
Notice that an autonomous differential equation is separable and that a solution can be found by
integrating
Since th
Exact Differential Equations
Consider the equation
f(x,y) = C
Taking the gradient we get
fx(x,y)i + fy(x,y)j = 0
We can write this equation in differential form as
fx(x,y)dx+ fy(x,y)dy = 0
Now divide by dx (we are not pretending to be rigorous here) to ge
Differences Between Linear and Nonlinear Equations
In this section we compare the answers to the two main questions in differential equations for
linear and nonlinear first order differential equations.
Recall that for a first order linear differential eq
Classification of Differential Equations
Recall that a differential equation is an equation (has an equal sign) that involves derivatives.
Just as biologists have a classification system for life, mathematicians have a classification
system for differenti
MATH 204
Differential Equations
Mon, Tues, Wed, and Thurs 1:00 to 2:35 AM
Room D108 (MW) B107 (TTh)
5 UNITS
Instructor: Larry Green
Phone Number
Office: 541-4660 Extension 341
Internet e-mail:.greenl@ltcc.edu
Home Page: http:/www.ltcc.edu/depts/math/
Your
Subspaces
Definition and Examples
Definition
Let V be a vector space and let S be a subset of V such that S is a vector space with
the same + and * from V. Then S is called a subspace of V.
Remark: Every vector space V contains at least two subspace, name
Change of Basis
Coordinates
Consider the vector v = (2,5,3) in R3. In writing these coordinates we mean
v = 2e1 + 5e2 + 3e3
Where
e1 = (1,0,0)
e2 = (0,1,0)
e3 = (0,0,1)
are the standard basis vectors. Sometimes we are interested in finding the coordinates
MATH 203 MIDTERM I
Part 1
Please work out each of the given problems without the use of a calculator. Credit will be based
on the steps that you show towards the final answer. Show your work.
Problem 1
Consider the matrix
A. Use the definition of the dete
Review of Some Linear Algebra
In this discussion, we expect some familiarity with matrices. For a review of the basics click
here. We will rely heavily on calculators and computers to work out the problems. Consider
some examples.
Example
Solve the system
Theory of Systems of Linear Differential Equations
It turns out that the theory of systems of linear differential equations resembles the theory of
higher order differential equations. This discussion will adopt the following notation. Consider
the system
Systems with Complex Eigenvalues
In the last section, we found that if
x' = Ax
is a homogeneous linear system of differential equations, and r is an eigenvalue with
eigenvector z, then
x = zert
is a solution. (Note that x and z are vectors.) In this discu
MATH 203
LINEAR ALGEBRA
Monday Wednesday and Friday 1:00 to 2:40 PM Room A211
Instructor: Larry Green
Phone Number
Office: 541-4660 Extension 341
Internet e-mail: DrLarryGreen@gmail.com
Home Page: http:/www.ltcc.edu/academics.asp?
scatID=5&catID=34" http:
Math 203 Practice Midterm 2
Please work out each of the given problems. Credit will be based on the steps towards the
final answer. Show your work.
Problem 1
Let L: R2 -> R3 be a linear transformation such that
L (1,4) = (1,-1,3) and
L (0,2) = (2,1,4)
Fin
Math 203 Practice Final Exam Printable Key
Please work out all of the following problems. Credit will be given based on the progress that
you make towards the final solution. Show your work. No calculators allowed for this page.
Problem 1
Let
Find an orth
Math 203 Practice Midterm 3
Please work out each of the given problems. Credit will be based on the steps towards the
final answer. Show your work. Do your work on your own paper.
Problem 1
A physicist has plotted the position of a projectile over time. B
Orthonormal Bases in Rn
Orthonormal Bases
We all understand what it means to talk about the point (4,2,1) in R3. Implied in this notation is
that the coordinates are with respect to the standard basis (1,0,0), (0,1,0), and (0,0,1). We learn
that to sketch
Homogeneous Systems
In this discussion we will investigate how to solve certain homogeneous systems of linear
differential equations. We will also look at a sketch of the solutions.
Example
Consider the system of differential equations
x' = x + y
y' = -2x
MATH 202 MIDTERM 1
Please work out five of the given six problems and indicate which problem you
are omitting. Credit will be based on the steps that you show towards the final
answer. Show your work.
PROBLEM 1 Please answer the following true or false. I
Surface Integrals
Surface Integrals for Parametric Surfaces
In the last section, we learned how to find the surface area for parametric surfaces. We cut the
region in the uv-plane into tiny rectangles and added up the area of the corresponding tiny
parall
Parametric Surfaces
Definition of a Parametric Surface
We have now seen many kinds of functions. When we talked about parametric curves, we
defined them as functions from R to R2 (plane curves) or R to R3 (space curves). Because each
of these has its doma