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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: