Schedule

Week 10: Review week

Core Topics

  1. Review of the practice problems for post-midterm material
  2. Comprehensive review

Some Practice Problems for the post-midterm material [Ch. 6, 7]

Week 9: Read Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.6

Core Topics

  1. Laplace transform: its definition, examples of Laplace transforms of elementary functions (constant, monomial, exponential, cos and sin, cosh and sinh) and their domains, inverse Laplace transform [Ch. 6.1]
  2. Basic properties of Laplace transform: linearity, $t$-domain scaling, $s$-domain derivative, $s$-domain integral, $t$-domain derivative, $t$-domain integral and convolution, $s$-domain shift
  3. Using Laplace transform to solve IVPs: partial fraction expansion [Ch. 6.2]
  4. Step (Heaviside) and impulse (Dirac Delta) functions [Ch. 6.3, 6.5]
  5. Solving IVPs with discontinuous forcing [Ch. 6.4]
  6. Convolution integrals: integro-DEs [Ch. 6.6]
  7. Solving matrix-vector IVPs using Laplace transform 

Week 8: Read Sections 7.8, 7.7, 7.9

Core Topics

  1. Some details on the general solution for matrix-vector linear homogeneous ODE with constant coefficients with repeated real eigenvalues [Ch. 7.8]

  2. Fundamental matrix and state transition matrix [Ch. 7.7]

  3. State transition matrix for the constant coefficient matrix-vector ODE: the matrix exponential

  4. State transition matrix for non-constant coefficient matrix-vector ODE: some solvable cases
  5. Solving non-homogeneous matrix-vector ODE [Ch. 7.9]

Week 7: Read Sections 7.5, 7.6, 7.8

Core Topics

Calculation details for the general solution of an example 4 x 4 system

  1. Solving matrix-vector linear homogeneous ODE with constant coefficients: all eigenvalues are real and unequal [Ch. 7.5]

  2. Solving matrix-vector linear homogeneous ODE with constant coefficients: some eigenvalues are complex conjugates [Ch. 7.6]

  3. Solving matrix-vector linear homogeneous ODE with constant coefficients: some eigenvalues are real equal [Ch. 7.8]  

Week 6: Read Sections 7.1, 7.4, 7.5

Lecture notes for Mon-May 07

Core Topics

  1. How to re-write an $n$-th order ODE as a system of $n$ first order ODEs [Section 7.1]

  2. Re-writing an $n$-th order linear ODE as a system of $n$ first order linear ODEs, or equivalently as a single $n$-th order matrix-vector ODE [Section 7.4]
  3. Theorems 7.4.1 (superposition principle), 7.4.2 (fundamental solution, general solution and Wronksian), 7.4.3 (Wronskian either always zero or always non-zero), 7.4.4 (at least one fundamental set), and 7.4.5 (real and complex part) 
  4. Homogeneous linear ODE with real constant coefficients: $\mathbf{x}^{\prime} = \mathbf{A}\mathbf{x}$, its equilibrium solution and general solution 

Week 5: Read Sections 3.6, 4.1, 4.2

Core Topics

  1. How to solve 2nd order linear non-homogeneous ODEs: method of variation of parameters [Section 3.6]

  2. Higher order linear ODE: general form, existence and uniqueness, Wronksian [Section 4.1]
  3. Higher order linear homogeneous ODE with constant coefficients: solution method [Section 4.2]
  4. Review for Mid-term [some practice problems]

Week 4: Read Sections 3.3, 3.4, 3.5

Core Topics

  1. How to solve 2nd order linear homogeneous constant coefficient ODEs: the case of complex conjugate roots [Section 3.3]

  2. How to solve 2nd order linear homogeneous constant coefficient ODEs: the case of real equal roots [Section 3.4]
  3. Relations between the solution of 2nd order linear non-homogeneous ODE and that of its homogeneous counterpart: Theorems 3.5.1 and 3.5.2
  4. How to solve 2nd order linear non-homogeneous ODEs: method of undetermined coefficients [Section 3.5]

Week 3: Read Sections 2.6, 3.1, 3.2

Core Topics

  1. How to solve exact ODEs (can solve some 1st order nonlinear ODEs) [Section 2.6]

  2.  Classification of 2nd order linear ODEs: homogeneous and non-homogeneous, further classification of homogeneous: constant coefficient and non-constant coefficient

  3.  How to solve 2nd order linear homogeneous constant coefficient ODEs: the case of real unequal roots [Section 3.1]

  4.  Existence and uniqueness of 2nd order linear ODEs [Section 3.2]

  5.   Theorems 3.2.1, 3.2.2, 3.2.3, 3.2.4, 3.2.5, 3.2.6, 3.2.7 and examples

  6.   Concept of Wronskian

  7.   General solution and fundamental solution

Week 2: Read Sections 2.1, 2.2, 2.4, 2.6

Core Topics

  1. Method of integrating factors (can solve all 1st order linear ODEs)

  2. Definition of separable ODEs

  3. How to solve separable ODEs (can solve some 1st order nonlinear ODEs)

  4. Theorem 2.4.1 on the existence and uniqueness of solutions to IVP of 1st order linear ODEs

  5. Theorem 2.4.2 on the existence and uniqueness of solutions to IVP of 1st order nonlinear ODEs

  6. Definition of 1st order exact ODEs

 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, linear/nonlinear, time varying/time invariant)
    4. The general form of linear differential equations
    5. How to solve first order, linear, time invariant ODEs
    6. How to solve first order, linear, time varying ODEs using integrating factors