MTH 2140 - Differential Equations

• Classify differential equations as linear or nonlinear and determine their order
• Analyze solutions to first-order differential equations graphically
• Verify if given functions are solutions of a differential equation, including identifying an interval of existence
• Solve separable first-order differential equations by the method of separation of variables, including finding singular solutions
• Solve first-order linear differential equations by the method of integrating factors
• Solve first-order initial value problems
• Model selected applications using first-order differential equations and interpret the results
• Approximate solutions with error estimation of first-order initial value problems using Euler’s method
• Use the Wronskian to determine if a set of solutions to a homogeneous linear differential equation are linearly independent
• Solve higher-order linear homogeneous differential equations by using the Superposition Principle and linear independence
• Solve higher-order homogeneous linear differential equations with constant coefficients using the auxiliary equation (characteristic equation)
• Solve higher-order linear nonhomogeneous differential equations by the method of undetermined coefficients
• Model selected applications using linear higher-order differential equations and interpret the results (such as spring-mass systems or LRC circuits)
• Determine the Laplace transform of selected elementary functions using the definition
• Determine the Laplace and inverse Laplace transform of selected elementary functions using tables and linearity
• Determine the Laplace and inverse Laplace transform of functions using the translation theorems
• Solve initial value problems of various orders using the method of the Laplace transform
• Perform basic matrix algebra operations, including addition, multiplication, and scalar multiplication
• Solve systems of linear equations using matrix methods
• Compute the determinant of a matrix using cofactor expansion
• Determine the eigenvalues and eigenvectors of a matrix
• Solve homogeneous linear systems with constant coefficients using the eigenvalue method (for nondefective matrices)
• Model selected applications using homogeneous linear systems and interpret the results

• Precalculus mathematics
• Partial fraction decomposition and basic polynomial factoring
• Limits, including L’Hôpital’s rule
• Differential and integral calculus of one variable

• Basic concepts and definitions
• Existence and uniqueness for first order initial value problems
• Direction fields of first order differential equations
• Solutions, including singular solutions, of first-order differential equations by the method of separation of variables
• Solutions of first-order linear differential equations by the method of integration factors
• Approximating solutions of first-order differential equations using Euler’s method
• The Wronskian of a set of functions
• Solutions of physical situations that can be modeled by first-order differential equations
• Solutions of higher order homogeneous differential equations with constant coefficients
• Solutions of non-homogeneous higher-order differential equations using the method of Undetermined Coefficients
• Solutions of physical situations that can be modeled by higher-order differential equations
• Definition of the Laplace transform and its properties
• The Laplace transform of elementary functions
• Inverse Laplace transform
• Translation theorems of the Laplace transform
• Solutions of linear differential equations with constant coefficients using the Laplace transform
• Basic matrix algebra operations and concepts
• Representing and solving algebraic linear systems with matrices
• Determinants of square matrices
• Linear independence
• Eigenvalues and eigenvectors of a matrix
• Solutions to systems of homogeneous linear systems with constant coefficients
• Solutions of physical situations that can be modeled by homogeneous linear systems