1
Introductory Material
a. First Order Differential Equations
i. Separation of Variables
1. [pic]
2. General Solutions – Implicit and Explicit
3. Initial Value Problems – Particular Solutions
ii. Linear Differential Equations
1. Find integrating factors and solve initial value problems
a. [pic]
[pic]
[pic]
b. [pic]
i. Solve characteristic equation [pic]
ii. Three Cases:
1. [pic][pic]
2. [pic]
3. [pic], [pic]
c. [pic]
i. [pic]
ii. [pic]
iii. [pic]
iv. If [pic] term is solution of homogeneous equation, multiply guess by additional powers of t.
d. [pic]
i. Solve characteristic equation [pic]
ii. Three Cases:
1. [pic]
2. [pic]
3. [pic], [pic]
1. Linear Systems of Differential Equations
a. Phase Plane and Phase Portraits
i. Find solutions from phase portrait.
ii. Equilibrium Solutions
iii. y vs x and x(t), y(t) plots
b. Eigenvalue Problems – Solve for eigenvalues and eigenfunctions.
c. Solution of systems – Use eigenvalues and eigenfunctions to construct solutions to systems.
d. Understand classification of Equilibrium Points and Connection to Phase Portraits, Eigenvalues and Solution Behavior.
e. Types: Stable/Unstable, Nodes, Foci, Centers, Degenerate Nodes, and Saddles.
f. Fundamental Matrix, Principal Matrix
2. Applications
a. Casting Second Order Equations as First Order Equations,
b. Interpreting Terms in a System
3. Methods of Integration
a. Substitution
a. Integration by parts [pic]
b. Trigonometric Integrals [pic],
c. Trigonometric Substitution
i. [pic] - tangent substitution
ii. [pic] - sine substitution
d. Integration using Partial Fraction Decomposition
4. Integrals you should be able to do (or similar ones)
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |[pic] |
5. Nonlinear Systems of Differential Equations
a. Autonomous Equations [pic]
i. Equilibrium solutions [pic]
ii. Classification (stable, unstable)
iii. Phase Lines, Bifurcation Diagrams,
iv. Types of bifurcation – saddle node, transcritical, pitchfork, Hopf
b. Nonlinear Systems
i. Linearization About Equilibrium (Fixed) Points
ii. Stability of Fixed Points
iii. Identifying Interesting Features of Nonlinear Systems
iv. Limit Cycles – Be able to identify
c. Phase space plots – Given a direction field, identify equilibria, limits cycles, etc.
d. Convert systems to polar form
6. Application Problems
a. Mixing Problems
b. Population Dynamics (Logistic, Lotka-Volterra, etc)
c. Masses plus springs
d. Nonlinear pendulum
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