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MA 354
MA 354 -- Differential Equations
- Introduction to Differential Equations
a) Modeling via Differential Equations: Population Growth
i) Unlimited Population Growth
ii) Logistic Population Models
iii) Predator - Prey Systems
b) Analytic, Qualitative, and Numerical Approaches
- First Order Differential Equations
a) Analytic Techniques
i) Separation of Variables
ii) Linear Differential Equations
iii) Change of Variables (optional)
b) Qualitative Techniques: Slope fields
c) Numerical Techniques: Euler's Method
d) Extstence and Uniqueness of Solutions
e) Bifurcations
- First Order Systems
a) Modeling via Systems
b) The Geometry of Systems
c) Analytic Methods for Special Systems
d) Euler's Method for Systems
- Linear Systems
a) Properties of Linear Systems and the Linearity Principle
b) Straight Line Solutions
c) Phase Planes for Linear Systems with Real Eigenvalues
d) Complex Eigenvalues
e) Special Cases: Repeated and Zero Eigenvalues
f) Second order Linear Equations
g) The Trace-Determinant Plane
- Forcing and Resonance
a) Forced Harmonic Oscillators
b) Sinusoidal Forcing
c) Undamped Forcing and Resonance
d) Amplitude and Phase of the Steady State
e) The Tacoma Narrows Bridge (optional)
- Laplace Transform
a) Laplace Transforms
b) Discontinuous Functions
c) Second Order Equations
d) Delta Functions and Impulse Forcing
e) Convolutions
- Numerical Methods (optional as a substitution for parts of Chapter V and Chapter VI)
a) Numerical Error in Euler's Method
b) Improving Euler's Method
c) The Runge-Kutta Method
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