Differential Equations
for Engineers:
the Essentials
Class 6 notes
Agenda: Class 6
Reviewing Homework Assignment 4
Lectures:
Qualitative Analysis
Qualitative Analysis: Example
Consider the ODE
We can solve this ODE, but even if we could not we could determine
important insights into the behavior of the solution by studying
Direction Field and Direction Line
Direction Field and Direction Line (2)
Direction field:
Draws vector representing value of .
),( ytf
),( ytf
Direction Field
Equilibrium Points
2−
4
5
6
),(
=
ytf
Direction line
Stable Equilibrium Points
4
5
6
Direction line for
)2)(1(),( −−= yyytf
0
Unstable Equilibrium Points
4
5
6
Direction line for
)2)(1(),( −−= yyytf
0
Solution to ODE as it depends
on initial condition
Solution:
Stability Revisited
Review: Prior Definition of System Stability
A linear system is said to be stable if its response to bounded initial
conditions is bounded.
A linear time-invariant system is damped if its response to bounded
initial conditions goes to zero as time goes to infinity.
Refined Definitions
Nonlinear systems can have equilibrium points other than the origin.
This causes us to refine our definitions:
Stability:
An equilibrium point is stable if ultimately stays near
when comes sufficiently close to .
E
y
)(ty
E
y
)(ty
E
y
E
y
Reading Assignment
Read:
Example Engineering Application of ODEs
Numerical Integration of ODEs
Using the computer to calculate a solution time-step by time-step
L
T
V
Variables in Aircraft Longitudinal Flight