Differential Equations

MADE: Differential Equations

MADE.A: First Order Differential Equations

MADE.B: Second and Higher Order Differential Equations

MADE.C: Systems of Differential Equations

MADE.D: Laplace Transforms

MADE.E: Series Solutions

MADE.F: Mathematical Connections

MADE.A.1: classify differential equation by type (i.e. ordinary/partial), order, and linearity

MADE.A.2: solve separable differential equations for general solutions and initial value problems

MADE.A.3: solve first order differential equations and initial value problems using integrating factors

MADE.A.4: use modeling software to solve more complex first order differential equations

MADE.A.5: draw direction fields containing solution curves for first order differential equations by hand and using modeling software

MADE.A.6: solve first order differential equations that apply to various real-world models including falling bodies, mixtures, population and the Logistic equation, continuously compounded interest, and other physics application

MADE.A.7: draw and interpret real world solutions to first order differential equations using modeling software

MADE.A.8: partially differentiate functions of multiple variables as it pertains to Exact Equations for first order differential equations

MADE.A.9: solve first order Exact Equations

MADE.B.10: determine whether a first or second order differential equation has a unique solution over a given interval by working with the Existence and Uniqueness Theorem

MADE.B.11: solve second order linear homogeneous and non-homogeneous equations by finding characteristic equations, using the method of undetermined coefficients and variation of parameters

MADE.B.12: solve second order differential equations that apply to various real-world models such as mass-spring systems, electric circuits, and economic growth

MADE.B.13: use vector function notation when discussing the structure of solutions sets for homogeneous systems as it pertains to the Wronskian

MADE.B.14: recognize the existence and uniqueness of solutions for second order linear differential equations and a fundamental set of solutions; verify that two solutions form a fundamental set by taking the Wronskian

MADE.B.15: recognize the structure of solution sets to higher order linear differential equations, the basic Existence and Uniqueness Theorem, and the generalization of the Wronskian for higher order equations.

MADE.B.16: solve higher order constant coefficient homogeneous equations

MADE.B.17: solve special case non-homogeneous second order ODE’s including Cauchy-Euler Equations

MADE.B.18: when given a solution to a non-homogeneous second order equation, find a second linearly dependent solution using reduction of order

MADE.B.19: recognize systems of differential equations and the basic existence and uniqueness results for the corresponding initial value problems

MADE.C.20: solve constant coefficient homogeneous systems using eigenvalues and eigenvectors; solve systems with real, distinct eigenvalues, as well as those with repeated and imaginary eigenvalues

MADE.C.21: draw Phase Portraits for solutions of homogeneous systems with constant coefficients by hand and using a modeling software

MADE.C.22: solve non-homogeneous systems of ODE’s using the method of undetermined coefficients and variation of parameters

MADE.C.23: determine which non-linear systems are locally linear, and identify the systems’ behavior about each critical point

MADE.C.24: plot locally linear systems by hand and using modeling software

MADE.C.25: apply various population models derived from locally linear systems including Lotka-Volterra, competition and cooperation models

MADE.D.26: use the integral definition to perform Laplace transforms for functions, such as, but not limited to polynomials, exponentials, and trigonometric functions; use a Laplace table to accurately and efficiently identify Laplace transforms, such as, but not limited to, the transforms for polynomials, exponentials, and trigonometric functions, and the product of these functions

MADE.D.27: perform inverse Laplace transforms using a variety of techniques, such as but not limited to, algebraic manipulation partial fraction decomposition

MADE.D.28: discuss the main properties of the Laplace transform which make it useful for solving initial value problems

MADE.D.29: solve first and second order differential equations using Laplace transforms that apply to real world fields such as Electrical and Mechanical Engineering

MADE.D.30: write piecewise functions as compositions of Step (Heaviside) functions

MADE.D.31: recognize the general uniqueness and existence of solutions for Step functions, and will use the Laplace transform to find solutions to Step functions.

MADE.D.32: find the Laplace transform of, the Dirac Delta function

MADE.D.33: solve linear systems of differential equations using Laplace transforms

MADE.E.34: review Power Series as an introduction to series solutions of differential equations

MADE.E.35: recognize ordinary points, recurrence relations, and changing indexes as it relates to series solutions to ODE’s

MADE.E.36: find series solutions to first and second order non-linear initial value problems

MADE.F.37: identify and describe the contribution of several key mathematicians and scientists to the field of differential equations