REQUIRED COURSE LEARNING OUTCOMES:

 1.  Recognize and classify differential equations.

 2.  Use graphical and numerical approaches to analyze solution curves.

 3.  Solve first and higher order linear, homogeneous and nonhomogeneous differential equations using classical techniques.

 4.  Solve first and higher order linear, homogeneous and linear nonhomogeneous differential equations using Laplace Transforms and power series.

 5.  Apply differential equations to solve various problems in the physical and natural sciences.

 6.  Define various introductory linear algebra concepts including vector spaces, subspace, basis, span, dimension, linear dependence/independence, linear transformations and determinants.

 7.  Solve systems of differential equations using eigenvalues and eigenvectors.

 RECOMMENDED TOPICAL OUTLINE:

 I.    Differential equations

  A.  Classification by type

  B.  Classification by order

  C.  Classification by linearity

 II.   Graphical and numerical approaches to analyze solution curves

  A.  Slope fields

  B.  Phase lines

  C.  Phase planes

  D.  Numerical methods including Euler's and Runge-Kutta methods

 III.  First and higher order linear, homogeneous and nonhomogeneous differential equations using classical techniques

  A.  Separation of variables

  B.  Integrating factor method

  C.  Method of undetermined coefficients

  D.  Method of variation of parameters

  E.  Methods of substitution such as reduction of order, y over x, etc.

  F.  Exact equations

  G.  Auxiliary equations including distinct roots, repeated roots, and imaginary roots

  H.  Linear independence

  I.  Wronskian determinants to prove linear independence

  J.  Existence and Uniqueness Theorem

 IV.   First- and higher-order linear homogeneous and linear nonhomogeneous differential equations using Laplace transforms and power series.

  A.  Laplace transforms of elementary functions

  B.  Laplace transforms of periodic functions and derivatives

  C.  Inverse of Laplace transforms

  D.  Power series solutions to differential equations

 V.    Differential equations to solve various problems in the physical and natural sciences.

  A.  Spring - mass systems

  B.  Growth and decay

  C.  Newton's Law of Cooling

 VI.   Various introductory linear algebra concepts including vector spaces, subspace, basis, span, dimension, linear dependence independence, linear transformations and determinants.

  A.  Matrix algebra

  B.  Inverse of matrices

  C.  Determinants of matrices

  D.  Vector spaces and subspaces

  E.  Basis and dimension

  F.  Linear transformations

  G.  Linear dependence/independence

  H.  Eigenvalues and eigenvectors

 VII.  Systems of differential equations using eigenvalues and eigenvectors.

         A.  Matrix forms for systems of differential equations

  B.  Distinct real, repeated, and complex eigenvalues

  C.  Linear systems with real and nonreal eigenvalues

  D.  Decoupling linear systems of differential equations

  E.   Classifying stability of equilibria in linear and nonlinear systems

 VIII. Matrix exponential

 IX.   Nonhomogeneous linear systems