MATHÂ 0033. Differential Equations and Linear Algebra
Units: 6
Prerequisite: Completion of MATH 31 with grade of "C" or better
Advisory: Completion of MATH 32 with grade of "C" or better strongly recommended
Hours: 108 lecture
First and second order ordinary differential equations, linear differential equations, numerical methods and series solutions, Laplace transforms, modeling and stability theory, systems of linear differential equations, matrices, determinants, vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors. (C-ID MATH 910S) (CSU, UC)
MATH 0033 - Differential Equations and Linear Algebra
http://catalog.sierracollege.edu/course-outlines/math-0033/
Catalog Description DESCRIPTION IS HERE: Prerequisite: Completion of MATH 31 with grade of "C" or better Advisory: Completion of MATH 32 with grade of "C" or better strongly recommended Hours: 108 lecture Description: First and second order ordinary differential equations, linear differential equations, numerical methods and series solutions, Laplace transforms, modeling and stability theory, systems of linear differential equations, matrices, determinants, vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors. (C-ID MATH 910S) (CSU, UC) Units 6 Lecture-Discussion 108 Laboratory By Arrangement Contact Hours 108 Outside of Class Hours Course Student Learning Outcomes Solve first and higher order ordinary and linear differential equations; using Laplace transformations, numerical, and series methods. Utilize theorems from linear algebra and use matrices to solve systems of equations, including differential equations. Utilize theorems from linear algebra to classify sets and mappings. Present clear, complete, accurate, and sufficiently detailed solutions to communicate reasoning and demonstrate the method of solving problems. Course Content Outline 1. First order differential equations including separable, homogeneous, exact, and linear; 2. Existence and uniqueness of solutions; 3. Applications of first order differential equations such as circuits, mixture problems, population modeling, orthogonal trajectories, and slope fields; 4. Second order and higher order linear differential equations; 5. Fundamental solutions, independence, Wronskian; 6. Nonhomogeneous equations; 7. Applications of higher order differential equations such as the harmonic oscillator and circuits; 8. Methods of solving differential equations including variation of parameters, Laplace transforms, and series solutions; 9. Systems of ordinary differential equations; 10. Techniques for solving systems of linear equations including Gaussian and Gauss-Jordan elimination and inverse matrices; 11. Matrix algebra, invertibility, and the transpose; 12. Relationship between coefficient matrix invertibility and solutions to a system of linear equations and the inverse matrices; 13. Special matrices: diagonal, triangular, and symmetric; 14. Determinants and their properties; 15. Vector algebra for Rn; 16. Real vector spaces and subspaces, linear independence, and basis and dimension of a vector space; 17. Matrix-generated spaces: row space, column space, null space, rank, nullity; 18. Change of basis; 19. Linear transformations, kernel and range, and inverse linear transformations; 20. Matrices of general linear transformations; 21. Eigenvalues, eigenvectors, eigenspace; 22. Diagonalization including orthogonal diagonalization of symmetric matrices; 23. Dot product, norm of a vector, angle between vectors, orthogonality of two vectors in Rn; 24. Orthogonal and orthonormal bases: Gram-Schmidt process; 25. Matrix exponential function for a system of differential equations; 26. Convolution integral; 27. Green's theorem; 28. Differential equations with forcing functions involving the unit step function and forcing functions involving the Dirac delta function; 29. Assess the need for the appropriate shifting theorems and apply when appropriate to solve a differential equation; 30. Cramer's rule; 31. Slope Fields including equilibrium solutions, Isoclines and concavity changes; 32. Inner Product Spaces including the norm of a vector and Cauchy-Schwarz Inequality; 33. Isomorphisms; 34. Quadratic and Jordan Canonical Forms; 35. Method of Undetermined Coefficients; and 36. LU Factorization. Course Objectives Course Objectives 1. Create and analyze mathematical models using ordinary differential equations; 2. Verify solutions of differential equations; 3. Identify the type of a given differential equation and select and apply the appropriate analytical technique for finding the solution of first order and selected higher order ordinary differential equations; 4. Apply the existence and uniqueness theorems for ordinary differential equations; 5. Find power series solutions to ordinary differential equations including Frobenius solutions; 6. Determine the Laplace Transform and inverse Laplace Transform of functions and use to solve differential equations with initial value conditions; 7. Solve Linear Systems of ordinary differential equations; 8. Find solutions of systems of equations using various methods appropriate to lower division linear algebra; 9. Use bases and orthonormal bases to solve problems in linear algebra; 10. Find the dimension of spaces such as those associated with matrices and linear transformations; 11. Find eigenvalues and eigenvectors and use them in applications; 12. Prove basic results in linear algebra using appropriate proof-writing techniques such as linear independence of vectors; properties of subspaces; linearity, injectivity and surjectivity of functions; and properties of eigenvectors and eigenvalues; 13. Verify that the axioms of a vector space, subspace, and inner product are satisfied or cannot be satisfied for a variety of sets including: n-dimensional space, polynomials, matrices, continuous and differentiable functions; 14. Examine Legendre and Bessel differential equations and their solutions; 15. Examine the phase plane for generating a qualitative representation of the solution to a system of nonlinear differential equations. Methods of Evaluation Objective Examinations Problem Solving Examinations Projects Reports Reading Assignments 1. Read in the textbook about the axioms of a vector space. Come to class prepared to discuss what it means to be a vector space and what it means not to be a vector space. 2. Read in your textbook (and research online) slope fields of the form D(y)=f(x,y) including isoclines, equilibrium solutions, and concavity. 3. Read in the textbook about the projection vector and examine how it is used to derive the formula in the Gram-Schmidt Process. Writing, Problem Solving or Performance 1. Sketch the slope field and some representative solution curves for the differential equation D(y)=y(y-1). 2. Use technology to graph the slope field and validate the algebraic calculations of isoclines, equilibrium solutions, and concavity. Other (Term projects, research papers, portfolios, etc.) Methods of Instruction Lecture/Discussion Distance Learning Other materials and-or supplies required of students that contribute to the cost of the course.
Course Identification Numbering System (C-ID)
...MATH 230 MATH 0032 MATH 900S MATH 0030 and MATH 0031 MATH 910S MATH 0033...
