MA 257 - Differential Equations and Linear Algebra

1. Effectively organize communications ensuring there is a clear introduction and conclusion, the content is well sequenced, and there are appropriate transitions. 

2. Clearly and completely state and describe a problem/issue. 

3. Complete work accurately, with attention to detail.

4. Solve simple differential equations, including Separable, Homogeneous, Exact, First-Order Linear, and Reducible to 1st Order. 

5. Apply differential equations to real-world problems.

6. Apply properties of Matrices and Determinants in connection with their computations and applications.

7. State definitions of Vector Space, Subspaces, Linear Dependence/Independence, Bases, Dimension, Linear Transformations and Differential Operators, and write correct proofs of statements associated with these.

8. Solve nth order Linear Differential Equations, especially those with constant coefficients.

9. Solve systems of differential equations.

10. Find Power Series solutions to differential equations.

11. Demonstrate numerical methods for solving differential equations, and be able to use technology to implement these methods.

12. Define Laplace Transforms; state, prove and use properties of Laplace Transforms and their inverses; and use Laplace Transforms to solve differential equations.  

II.  Matrices and Determinants

     A. Systems of Linear Equations

     B. Matrices and Vectors

     C. Matrix Operations

     D. Determinants

     E. Cramer’s Rule

     F. The Inverse of a Matrix

III. Vector Spaces and Linear Transformations

     A. Vector Spaces

     B. Subspaces and Spanning Sets

     C. Linear Independence of Vectors

     D. The Wronskian

     E. Basis and Dimension of a Vector Space

     F. Linear Transformations

     G. Matrix Representation of Linear Transformations

     H. Kernel and Range

      I. Differential Operators

     J. Multiplication of Differential Operators

     K. Eigenvalues and Eigenvectors, including Complex Numbers Cases    

IV. Linear Differential Equations

     A. Definition of nth Order Linear Differential Equations

     B. Solutions of Equations with Constant Coefficients

     C. Non-Constant Coefficients: Cauchy-Euler Equations

     D. General Solution of Non-Homogeneous Equations

     E. Method of Undetermined Coefficients

     F. Variation of Parameters

     G. Applications

V.  Systems of Differential Equations

     A. Definition of First Order Systems

     B. Solution of Systems by Elimination or Substitution

     C. Representation of Systems by Matrices

     D. Solution of Systems by Eigenvectors

     E. Non-Homogeneous Linear Systems

     F. Applications

VI. Series Solutions

     A. Power Series

     B. Taylor Series

     C. Operations with Series including re-indexing

     D. Series Solutions: Ordinary Points

     E. Series Solutions: Singular Points

VII. Numerical Methods

     A. Euler Method

     B. Runge-Kutta Methods

VIII. Laplace Transforms

     A. Definition of Laplace Transform

     B. Computing Laplace Transforms

     C. Properties of Laplace Transform

     D. Computing Inverse Laplace Transforms

     E. Solving Differential Equations using Laplace Transforms