Week 10: Read Section 6.3, 6.4 and review all topics including those coverred before the midterm exam
Here is a set of formulas that will be provided to you in the final exam
Some practics problems
Core Topics
- How to use partial fractions to find inverse Laplace transform.
- How to solve initial value problems using Laplace transform.
- Definition of Unit Step Function.
- Laplace transform of uc(t)f(t-c) (Theorem 6.3.1).
- How to use unit step function to represent discontinuous functions.
- How to solve linear ODEs with discontinuous forcing functions.
Quiz in week 9 during lab sections. If you are in Monday section, you will take the quiz on the Monday of week 10.
Week 9: Read Section 6.1, and 6.2
Core Topics
- An engineering application of linear differential equations (click here for the slides)
- Definition of Laplace transform (Equation 4 in Section 6.1).
- Existence result for Laplace transform (Theorem 6.1.2).
- Linear property of Laplace transform and inverse Laplace transform.
- Laplace transform of some elementary functions.
- Theorem 6.2.1 and Corollary 6.2.2
- Laplace transform of derivative and higher order derivatives of a function (Theorem 6.2.1 & 6.2.2).
- How to use partial fractions to find inverse Laplace transform.
- How to solve initial value problems using Laplace transform.
Homework 9
Homework 9 — Solutions
Week 8: Read Section 7.6, 7.7, 7.8 and 7.9
Core Topics
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Construction of fundamental solutions for complex eigenvalues.
- Definition of matrix exponential function and its properties.
- How to compute matrix exponential function.
- How to find general solution of x'=Ax using matrix exponential function.
- How to solve initial value problems for homogeneous linear system using matrix exponential function.
- How to solve non-homogeneous system of linear ODE, x'=Ax+G(t).
Homework 8
Homework 8 — Solutions
Week 7: Read Section 7.4, 7.5 And 7.6
Core Topics
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Basic theory of systems of first order linear ODEs.
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How to construct general solution for x'=Ax, where A has a full set of linearly independent eigenvectors.
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How to diagonalize a matrix and use diagonalization to solve x'=Ax.
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How to find general solution of x'=Ax where A has a full set of linearly independent real eigenvectors.
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Construction of fundamental solution when A has an eigenvalue whose geometric multiplicity is smaller than its algebraic multiplicity.
Homework 7
Homework 7 — Solutions
Week 6: Read Section 7.1, 7.2, 7.3 and 7.4
Core Topics
- Definition of systems of 1st order ODEs.
- How to transfer a higher order ODE to a system of 1st order OED.
- Review linear algebra including the following topics.
- Concepts of linearly independent vectors, eigenvalue and eigenvector.
- 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)
- Basic theory of systems of first order linear ODEs.
Homework 6
Homework 6 — Solutions
Midterm: May 08, 2017
Here Is a set of formulas that will be provided to you in the midterm
Week 5: Read Section 3.8, 4.1 And 4.2
Core Topics
- How to analyze forced vibrations (Section 3.8).
- Theorem 4.1.1 on the existence of the solution.
- Theorem 4.1.2 on the structure of the general solution for homogeneous ODE.
- Definition of linear independency of funcitons, and its relation to fundamental set of solutions (Theorem 4.1.3).
- General solution to nonhomogeneous linear ODEs.
- How to solve homogeneous linear ODEs with constant coefficients (Section 4.2)
Week 4: Read Section 3.5, 3.6, and 3.7
Core Topics
- Theorem 3.5.2 on the structure of solutions to 2nd order nonhomogeneous linear ODEs.
- How to find a particular solution to 2nd order nonhomogeneous linear OEDs using Varition of Parameters.
- Theorem 3.6.1 and its proof.
- 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.
- Mechanical and Electrical Vibrations (Section 3.7).
Homework 4
Homework 4 — Solutions
Week 3: Read Section 3.1, 3.2, 3.3 And 3.4
Core Topics
- Definition of Wronskian determinant.
- 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.
- Definition of fundamental set of solutions and general solution.
- Theorem 3.2.5 on the existence of fundamental set of solutions.
- The concept of linear independent funcitons.
- The concept of characteristic equation for 2nd order linear OEDs with constant coefficients.
- How to find fundamental set of solutions to 2nd order homogeneous linear OEDs with constant coefficients.
- How to solve IVP of 2nd order homogeneous linear OEDs with constant coefficients.
Homework 3
Homework 3 — Solutions
Week 2: Read Section 2.1, 2.2, 2.3, 2.4, 3.1 And 3.2
Core Topics
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Method of integrating factors.
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Concept of Interval of Definition for the solution to initial value problems (IVP).
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Theorem 2.4.1 on the existence and uniqueness of solutions to IVP of 1st order linear ODEs.
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Definition of separable equations.
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How to solve separable equations.
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How to find interval of definition for the solution to IVP of separable equations.
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Theorem 2.4.2 on the existence and uniqueness of solutions to IVP of 1st order nonlinear ODEs.
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Definition of 2nd order linear homogeneous/nonhomogeneous equations.
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Theorem 3.2.1 on the existence and uniqueness of solutions to IVP of linear 2nd order ODEs.
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Theorem 3.2.2 Principle of Superposition for homogeneous linear 2nd order ODES.
Homework 2
Homework 2 — Solutions
Week 1: Read Chapter 1 and Section 2.1 (Method of Integrating Factors)
Core Topics
- Definitions of differential equations, initial condition, initial value problem, and general solution.
- How to verify a given function is a solution to a deferential equation.
- How to classify differential equations (ODE/PDE, order, dimension, linear/nonlinear, time varying/time invariant)
- The general format of linear differential equations.
- How to solve first order, linear, time invariant differential equations.
- How to solve first order, linear, time varying differential equations using integrating factors.
Homework 1
Homework 1 — Solutions