Download Damien High School Mathematics and Computer Science

Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Differential Equations
“A” in AP Calculus BC and a score of “5” on the AP Calculus BC exam
Yes – Category C
Year
Brief Course Description
This course provides an introduction to ordinary differential equations with an emphasis on applications. Topics include first-order, linear higher-order,
and systems of differential equations; numerical methods; series solutions; eigenvalues and eigenvectors; Laplace transforms; and Fourier series. Upon
completion, students should be able to use differential equations to model physical phenomena, solve the equations, and use the solutions to analyze the
phenomena.
Assigned Textbook(s)
Supplemental Material(s)
A First Course in Differential Equations with Modeling Applications, 9th ed., by Dennis Zill
Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
Tests
ISOs Addressed
Overview of Course / Skill Outcomes
This section serves as a precursor to the Curriculum Map and, as such, should briefly describe the various units (major content chunks) that comprise the course as well as the skills / techniques
necessary to be successful in the course.
Major Content Outcomes
Students will be able to solve first-order differential equations.
Students will be able to solve higher-order differential equations.
Students will be able to use matrices.
Students will be able to use the properties of the Laplace Transform and Inverse Laplace.
Students will be able to analyze, interpret, and solve applications of ODE’s
Students will be able to solve systems of linear first order differential equations.
I.
II.
III.
IV.
First Order Differential Equations
A. Initial Value Problems
B. Differential Equations as Mathematical Models
C. Solution Curves Without a Solution
D. Direction Fields and Autonomous First Order D.E.
E. Separable Variables
F. Linear Equations
G. Exact Equations
Higher Order Differential Equations
A. Preliminary Theory for Linear Equations
1. Initial Value and Boundary Value Problems
2. Homogeneous Equations
3. Nonhomogeneous Equations
B. Reduction of Order
C. Homogeneous Linear Equations with Constant Coefficients
D. Variation of Parameters
E. Cauchy Euler Equations
The Laplace Transform
A. Definition of Laplace Transforms
B. Inverse Transforms and Transforms of Derivatives
C. Operational Properties
1. Translation on the s-axis
2. Translation on the t-axis
3. Derivatives of a Transform
4. Transforms of Integrals
5. Transform of a Periodic Function
D. The Dirac-Delta Function
Matrices
A. Operations with Matrices
Major Skill Outcomes (include Labs???)
Students will learn what an Ordinary Differential Equation (ODE) is, how to classify them,
what initial value problems are, & what constitutes a solution.
Students will learn to visualize and manipulate ODEs in graphical, numerical, and symbolic
form.
Students will understand the concepts of the existence and uniqueness of solutions.
Students will learn to work with matrices and apply them when dealing with determinants,
Cramer’s Rule, and Gauss Jordan Elimination.
Students will recognize certain basic types of first order and higher order ODEs for which exact
solutions may be obtained and will solve them using the corresponding methods.
Students will be introduced to the concept of the Laplace transform and will apply the
properties to solve linear ODE’s.
Students will be introduced to system of linear first order ODE’s and discuss graphical,
numerical, and analytical solution methods
Students will work with a variety of applications, using appropriate models, and will analyze the
validity of the solutions obtained.
Unit 1
Students will be able to identify differential equations by various criteria.
Students will be able to solve separable differential equations.
Students will be able to solve first order linear differential equations.
Students will be able to solve exact equations.
Students will be able to solve homogeneous differential equations.
Unit 2
Students will recognize and solve initial value problems, boundary value problems,
homogeneous, & non-homogeneous differential equations.
Students will be able to find a second solution of a differential equation from a known solution,
using reduction of order.
Students will be able to solve homogeneous and non-homogeneous linear equations with
constant coefficients.
Students will learn how to solve Cauchy Euler equations.
Students will be able to solve certain types of nonlinear differential equations.
Unit 3
Students will be able to find the Laplace (and inverse Laplace) transform of functions and
V.
B. Derivatives and Integrals of Matrices of Functions
C. Row Echelon Form and Gauss-Jordan Elimination
D. Eigenvalues and Eigenvectors
Systems of Linear First-Order Differential Equations
A. Preliminary Theory for Linear Systems
B. Homogeneous Linear Systems
1. Distinct Real, Repeated, and Complex Eigenvalues
C. Nonhomogeneous Linear Systems
1. Undetermined Coefficients
2. Variation of Parameters
D. Matrix Exponential
derivatives by using the definition and formulas.
Students will be able to use Laplace transforms to solve differential equations with initial
conditions.
Students will be able to translate on the s-axis and t-axis.
Students will be able to find derivatives of transforms and transforms of integrals.
Students will be able to find the transform of a periodic function.
Students will be able to solve differential equations having a Dirac Delta function.
Unit 4
Students will be able to find when two matrices are equal and apply operations involving
matrices.
Students will be able to find the derivative and integral of a matrix of functions.
Students will be able to solve a system of equations by Gaussian and Gauss-Jordan elimination.
Students will be able to find eigenvalues and eigenvectors of a matrix.
Unit 5
Students will be able to solve a system of homogeneous linear systems.
Students will be able to solve a system of nonhomogeneous linear systems.
Students will be able to use the matrix exponential.
Differential Equations
Unit 1 – What are First-order Differential Equations?
Content Outcomes
Students will be able to identify differential
equations by various criteria.
Essential Questions
How can we identify the order of an ordinary
differential equation and determine whether it is
linear or nonlinear?
Key Concepts
Definition of separable equations
Standards Addressed
California:
Common Core:
How can we sketch a slope field for a first-order
differential equation as well as solution curves on
the slope field?
How do we apply the Existence-Uniqueness
Theorem for first-order differential equations.
Students will be able to solve separable differential
equations.
How do we identify a separable first-order equation
and find a family of solutions or a particular
solution?
Students will be able to solve first order linear
differential equations.
How do we identify a first-order linear equation and
find the general solution using an integrating factor?
Definition of linear equations
California:
Common Core:
Students will be able to solve exact equations.
What is an exact equation?
Definition of exact equations
How do we identify an exact differential equation
and find a family of solutions?
California:
Common Core:
How do we solve initial-value problems involving
first-order separable, linear, and exact equations?
Students will be able to solve homogeneous
differential equations.
What is a homogeneous equation?
Definition of homogeneous equations
How do you solve a homogeneous equation?
Differential Equations
Unit 2 – What are Higher Order Equations?
California:
Common Core:
Content Outcomes
Students will recognize and solve initial value
problems, boundary value problems, homogeneous,
& non-homogeneous differential equations.
Essential Questions
What is an initial value problem?
What is a boundary value problem?
What is a higher order homogeneous equation?
Students will be able to find a second solution of a
differential equation from a known solution, using
reduction of order.
Students will be able to solve homogeneous and
non-homogeneous linear equations with constant
coefficients.
.
What is a nonhomogeneous equation?
How do we find a second solution if we know one
solution of a differential equation?
Key Concepts
Definition of linear dependence and independence
Definition of Wronskian
Definition of a fundamental set of solutions
Definition of complementary and particular
solutions.
Standards Addressed
California:
Formula for finding a second solution
California:
Common Core:
Common Core:
What is an auxiliary equation?
Variation of Parameters technique
How do we solve homogeneous linear equations
with constant coefficients?
California:
Common Core:
When do we use the method of Variation of
Parameters and how do we apply the process?
Students will learn how to solve Cauchy Euler
equations.
What is the form of Cauchy Euler Equations?
Definition of Cauchy Euler equations
How do we solve a Cauchy Euler Equation?
Students will be able to solve certain types of
nonlinear differential equations.
How do we solve nonlinear differential equations
where the dependent variable x or independent
variable y is missing?
California:
Common Core:
Reduction of order for nonlinear differential
equations
California:
Common Core:
Content Outcomes
Students will be able to find the Laplace (and
inverse Laplace) transform of functions and
derivatives by using the definition and formulas.
Differential Equations
Unit 3 – What is a Laplace Transform?
Essential Questions
Key Concepts
What is the definition of the Laplace transform?
Definition of the Laplace transform
Formulas of Laplace transforms
How do we find an inverse Laplace transform?
Formulas of inverse Laplace transforms
Formula for Laplace transforms of derivatives
How do we find the Laplace transform of
derivatives?
Students will be able to use Laplace transforms to
solve differential equations with initial conditions.
How do we use the Laplace transform to solve
differential equations and initial value problems?
Procedure for applying the Laplace transform and
inverse Laplace transform to solve an ODE
Standards Addressed
California:
Common Core:
California:
Common Core:
Students will be able to translate on the s-axis and
t-axis.
How do we translate on the s-axis?
First Translation Theorem
Second Translation Theorem
How do you convert a piecewise function into unit
step functions?
California:
Common Core:
How do we translate on the t-axis?
Students will be able to find derivatives of
transforms and transforms of integrals.
How do we find the derivative of transforms?
How do we find the transforms of integrals?
Students will be able to find the transform of a
periodic function.
How do we find the transform of a periodic
function?
Derivatives of transforms
Convolution Theorem
Transforms of integrals
California:
Transform of a periodic functions
California:
Common Core:
Common Core:
Students will be able to solve differential equations
having a Dirac Delta function.
What is the Dirac Delta function and how do we
solve differential equations involving them?
Transform of the Dirac Delta function
California:
Common Core:
Content Outcomes
Students will be able to find when two
matrices are equal and apply operations
involving matrices.
Students will be able to find the derivative and
integral of a matrix of functions.
Students will be able to solve a system of
equations by Gaussian and Gauss-Jordan
elimination.
Differential Equations
Unit 4 – What are Matrices?
Essential Questions
Key Concepts
When are 2 matrices equal?
Dimensions of a matrix
Equality of matrices
When and how can we find the sum, difference, Sums and products of matrices
and product of two matrices?
Inverse of a matrix
Determinant of a square matrix
How do we find the transpose and inverse of a Transpose of a matrix
matrix?
How do we differentiate a matrix of functions?
Formula for derivative of a matrix
Formula for integral of a matrix
How do we integrate a matrix of functions?
What row operations can be applied to a matrix to
maintain its equivalency?
Definition of augmented matrix
Definition of row echelon/ reduced row echelon
form
How do we solve a system of linear equations by
applying elementary row operations?
What are eigenvalues and eigenvectors?
How can we find the eigenvectors and eigenvalues
of a matrix?
Common Core:
California:
Common Core:
How do we find an inverse matrix by using
Gaussian elimination?
Students will be able to find eigenvalues and
eigenvectors of a matrix.
Standards Addressed
California:
Definition of eigenvalue
Definition of eigenvector
California:
Common Core:
Content Outcomes
Students will be able to solve a system of
homogeneous linear systems.
Students will be able to solve a system of
nonhomogeneous linear systems.
Differential Equations
Unit 5 – How do you Solve Systems of Linear First-order Differential Equations?
Essential Questions
Key Concepts
What is the definition of a system of linear first- Definition of a system of linear first-order
order differential equations?
differential equations
How do you solve a system with distinct real
eigenvalues?
Process of solving systems of homogeneous linear
systems
Standards Addressed
California:
Common Core:
How do you solve a system with repeated
eigenvalues?
Students will be able to solve a system of
nonhomogeneous linear systems.
Students will be able to use the matrix
exponential.
How do you solve a system with complex
eigenvalues?
How do you solve nonhomogeneous linear systems?
Solving non homogeneous linear systems by
Variation of Parameters
What is Variation of Parameters?
What is the matrix exponential?
California:
Common Core:
Definition of the matrix exponential
California:
Common Core: