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Math 2552 DDifferentialEquationsGaTech Spring 2017
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What's a differential equation?
any relation between a function and it's derivatives
dependent variable (time)
independent variable
derivatives of dependent variable
Why?We usually know local rules for a system.Use them to compute the global futures!
Brannan & Boyce
Course Goals:
Setting Up Equations
Finding Exact Solutions
Finding Numerical Solutions
Analyzing Equilibria
relating rates
analyze local & short term bevahior
dynamical systems
long term behavior
find most likelyoutcomes
exact solutions are usually impossible
strategy depends on equation type
usually guess and check
equation types
function space transforms
linear equations
a computer compute approximate solutions
sensitivity to initial conditions:CHAOS
1.1: The Name of the Wind
diffeq gives tangent vectors
solution curves follow tangents
diffeqs have MANY solutions
depends on the INITIAL CONDITIONS
SLOPE/DIRECTION FIELDS
GeoGebra
Bluffton
Draw a slope field
Sketch the solution with initial condition
Determine its long term behavior.
Jargon
OrderLinearHomogenousAutonomousOrdinary/Partial
refers to the highest derivative involved
linear dependence between variables and derivatives
every term dependson the dependent variable
the diffeq doesn't dependon the independent variable
a parrtial diffeq relates derivativeswith respect to differnt variables
third
Polling