Notes

Here are my own notes for CME 102. These roughly follow the material in Prof. Hung Le’s iteration of the course. Since Winter 2015, I have been involved with Prof. Le’s CME 102 courses, and over time have developed my own explanations or methods for working through problems. Differential equations in engineering are very computationally-intensive, and consequently these notes tend to emphasize algorithms for deriving a solution, although I do try to inject some theory when appropriate or useful. For my usual disclaimer, these aren’t here to replace the course reader and especially not lecture attendance, but hopefully they are some help.

• Front Matter: what CME 102 is all about, how should you study for this class, etc.
• Introduction: what are ODE, and why should we study them?
• Classifying ODEs: definition of order, homogeneity, and linearity
• Separation of Variables: overview, examples
• Equilibria and Stability: direction fields, phase space, stable and unstable solutions
• Linear First Order ODE: classification, solution method, Bernoulli equation
• Existence and Uniqueness: the existence and uniqueness theorems, examples
• Eigenvectors and ODEs: eigenvectors/eigenvalues, applications to systems of ODEs
• Numerical Methods 1: Euler methods, accuracy, stability, ode45
• Nonlinear ODEs: missing y and missing x methods, missing x and y equations
• Numerical Methods 2: higher order schemes, higher order and systems of ODE
• Second Order ODE 1: overview, reduction of order, characteristic equation
• Second Order ODE 2: variation of parameters, undetermined coefficients
• Numerical Methods 3: direct method, multistep methods, shooting method
• Mass-Spring Systems: free and forced mass-spring systems
• Laplace Transforms: important theorems, discontinuous functions, solving ODEs
• Power Series: ratio test, power series definition, solutions to ODEs

Section Worksheets

Below is a list of worksheets and other review sheets I created for CME 102 over several quarters. Caveat emptor: these are hopefully mistake-free, but there may be errors (if you find any, please let me know so I can make necessary corrections).

Worksheet	Description
Week 2 (Solution)	Linearity, separation of variables, direction fields, equilibria and stability, MATLAB
Week 3 (Solution)	Linear first order ODE, numerical accuracy, systems of linear equations review
Week 4 (Solution)	Existence and uniqueness, numerical stability, ODEs and the eigenvalue problem
Week 5 (Solution)	Nonlinear ODE, second order linear ODE, Cauchy-Euler equations, ode45, adaptive step solvers
Week 6 (Solution)	Undetermined coefficients, variation of parameters, amplification factor and stability
Week 7 (Solution)	Undetermined coefficients, variation of parameters, direct method
Week 8 (Solution)	Beats and resonance, mass spring systems, Laplace transforms
Week 9 (Solution)	Laplace transform, delta and step functions
Week 10 (Solution)	Power series
MATLAB Review	Powerpoint slides used for MATLAB review session in CME 102. Covers all relevant MATLAB from CME 100 necessary for CME 102.
Midterm 2 Review Problems (Solution)	Review problems for the topics for the second midterm: second order ODEs, mass-spring systems, higher order ODEs, numerical methods.
Laplace Transform Review (Solution)	Laplace transform review problems. Covers all topics including all forward/inverse transform theorems and discrete/discontinuous inputs.
Power Series Review (Solution)	Example problems for power series solution of ODEs.
Final Review Problems (Solution)	Review problems covering most important topics from the quarter. Included are several review problems for direct method.