Schedule and Homework Assignments

Week 10: Read 6.4 and review all topics including those coverred before the midterm exam

Here is a set of formulas that will be provided during the final exam

Some practice problems

Solution of Homework 9

Core Topics

  1. How to use unit step function to represent discontinuous functions.
  2. How to solve linear ODEs with discontinuous forcing functions.

Week 9: Read Section 6.1, 6.2 and 6.3

Quiz 4 on system of linear equations will be given in Week 9 during lab sections.

Core Topics
  1. Definition of Laplace transform (Equation 4 in Section 6.1).
  2. Existence result for Laplace transform (Theorem 6.1.2).
  3. Linear property of Laplace transform and inverse Laplace transform.
  4. Laplace transform of some elementary functions.
  5. Theorem 6.2.1 and Corollary 6.2.2
  6. Laplace transform of derivative and higher order derivatives of a function (Theorem 6.2.1 & 6.2.2).
  7. How to use partial fractions to find inverse Laplace transform.
  8. How to solve initial value problems using Laplace transform.
  9. Definition of Unit Step Function.
  10. Laplace transform of uc(t)f(t-c) (Theorem 6.3.1).
  11. How to use unit step function to represent discontinuous functions.

Homework 9

Week 8: Read Section 7.6, 7.7, 7.8 and 7.9

Core Topics

  1. Construction of fundamental solution when A has an eigenvalue whose geometric multiplicity is smaller than its algebraic multiplicity.
  2. Definition of matrix exponential function and its properties.
  3. How to compute matrix exponential function.
  4. How to find general solution of x'=Ax using matrix exponential function. 
  5. How to solve initial value problems for homogeneous linear system using matrix exponential function.
  6. How to solve non-homogeneous system of linear ODE, x'=Ax+G(t).

Homework 8

Week 7: Read Section 7.4, 7.5 and 7.6

Quiz 3 on higher order linear ODEs and eigenvalue/eigenvector computation will be given in Week 7 during lab sections.

Core Topics
  1. Basic theory of systems of first order linear ODEs.
  2. How to construct general solution for x'=Ax, where A has a full set of linearly independent eigenvectors.
  3. How to diagonalize a matrix and use diagonalization to solve x'=Ax.
  4. How to find general solution of x'=Ax where A has a full set of linearly independent real eigenvectors. 
  5. Construction of fundamental solutions for complex eigenvalues. 
  6. Construction of fundamental solution when A has an eigenvalue whose geometric multiplicity is smaller than its algebraic multiplicity.

Homework 7

Week 6: Read Section 7.1, 7.2, 7.3 and 7.4

Core Topics

  1. Definition of systems of 1st order ODEs. 
  2. How to transfer a higher order ODE to a system of 1st order OED.
  3. Review linear algebra including the following topics.
      • Concepts of linearly independent vectors, eigenvalue and eigenvector.
      • How to verify linear dependency of a set of vectors.
      • How to compute eigenvalue and eigenvector.
      • Concepts of algebraic multiplicity and geometric multiplicity.
      • Properties of diagonalizable matrix.
      • Definition of matrix functions and their properties (see Section 7.2)
  4. Basic theory of systems of first order linear ODEs.

Homework 6

Midterm Exam: Monday, 05/02/2016 (AMS 20A students are not required to take the midterm)

Practice Problems for Midterm Exam
Here is a set of formulas that will be provided to you during the midterm

Week 5: Read Section 3.7, 3.8, 4.1 and 4.2

Core Topics

  1. How to analyze mechanical and electrical vibrations (Section 3.7 & 3.8).
  2. Theorem 4.1.1 on the existence of the solution. 
  3. Theorem 4.1.2 on the structure of the general solution for homogeneous ODE.
  4. Definition of linear independency of funcitons, and its relation to fundamental set of solutions (Theorem 4.1.3).
  5. General solution to nonhomogeneous linear ODEs.
  6. How to solve homogeneous linear ODEs with constant coefficients (Section 4.2)

Week 4: Read Section 3.5, 3.6 and 3.7

Quiz 2 on 2nd order homogeneous linear ODE will be given in Week 4 during lab sections.

Core Topics
  1. Theorem 3.5.2 on the structure of solutions to 2nd order nonhomogeneous linear ODEs.
  2. How to find a particular solution to 2nd order nonhomogeneous linear OEDs using Varition of Parameters.
  3. Theorem 3.6.1 and its proof.
  4. Understand Table 3.5.1 and know how to use it to find a particular solution to 2nd order nonhomogeneous linear OEDs with constant coefficients using Method of Undetermined Coefficients.
  5. Mechanical and Electrical Vibrations (Section 3.7).

Homework 4

Week 3: Read Section 3.1, 3.2, 3.3 and 3.4

Quiz 1 will be given in Week 3 during lab sections.

Core Topics
  1. Definition of Wronskian determinant.
  2. Theorem 3.2.3 and Theorem 3.2.4 on Wronskian determinant and the structure of solutions to IVP of 2nd order homogeneous linear ODEs.
  3. Definition of fundamental set of solutions and general solution.
  4. Theorem 3.2.5 on the existence of fundamental set of solutions.
  5. The concept of linear independent funcitons.
  6. The concept of characteristic equation for 2nd order linear OEDs with constant coefficients.
  7. How to find fundamental set of solutions to 2nd order homogeneous linear OEDs with constant coefficients.
  8. How to solve IVP of 2nd order homogeneous linear OEDs with constant coefficients.

Homework 3

Week 2: Read Section 2.1, 2.2, 2.3, 2.4, 3.1 And 3.2

Core Topics
  1. Method of integrating factors.
  2. Concept of Interval of Definition for the solution to initial value problems (IVP).
  3. Theorem 2.4.1 on the existence and uniqueness of solutions to IVP of 1st order linear ODEs.
  4. Definition of separable equations.
  5. How to solve separable equations.
  6. How to find interval of definition for the solution to IVP of separable equations.
  7. Theorem 2.4.2 on the existence and uniqueness of solutions to IVP of 1st order nonlinear ODEs.
  8. Definition of 2nd order linear homogeneous/nonhomogeneous equations.
  9. Theorem 3.2.1 on the existence and uniqueness of solutions to IVP of linear 2nd order ODEs.
  10. Theorem 3.2.2 Principle of Superposition for homogeneous linear 2nd order ODES.

Homework 2

Week 1: Read Chapter 1 And Section 2.1 (Method Of Integrating Factors)

Core Topics
  1. Definitions of differential equations, initial condition, initial value problem, and general solution.
  2. How to verify a given function is a solution to a deferential equation.
  3. How to classify differential equations (ODE/PDE, order, dimension, linear/nonlinear, time varying/time invariant)
  4. The general format of linear differential equations.
  5. How to solve first order, linear, time invariant differential equations.
  6. How to solve first order, linear, time varying differential equations using integrating factors.

Homework 1