Dynamic Systems and Ordinary Differential Equations Maple

1
Dynamic Systems and Ordinary Differential
Equations - Maple

• What is an ordinary differential equation?
• Constructing models that use ODEs
• Mathematical solution of a simple ODE
• Symbolic computation solution of various ODEs

2
What is an ODE?
First order ODEs relate the first derivative of
a function (time rate of change) with the
function itself. The ODE is first order if only
the first derivative of the function is included.
Exponential growth or decay is governed by this
simple ODE. Plug in y(t) to check.
3
Solving an ODE
Answer the question what function can we
construct such that it has the relationship to
its derivative that the ODE expresses?
Solving an ODE by hand requires lots of
mathematical intuition and experience.
4
An engineering process can often be described
using ODEs

• Radioactive decay
• Chemical processes
• Controls of almost anything
• Population dynamics
• Diffusion of almost anything
• Transient electrical circuits

5
ODEs and Linear Systems

• ODEs express the relationship between different
  functions and their derivatives.
• ODEs often represent dynamic systems that vary
  in time.

• Linear systems express the relationship between
  different variables.
• Linear systems usually represent static systems,
  that is not varying in time.

6
How to solve simple first order ODE
7
Plot y(t) for positive and negative a
y(t)10e2t
y(t)10e-2t
y(t) either grows or decays very rapidly as t
increases. This is exponential growth or decay.
A constant value of y(t) demands that a0.
8
A model for population dynamics
9
Solve population ODE with Maple
In particular, choose the birth-death rate to be
2. Thus the birth rate exceeds the death rate. We
would expect the population to rise. How fast
does it rise? Solve the ODE and find out.
10
Maple commands and output
11
Plot solution function
12
A different model ofpopulation dynamics
13
Solve more complex population model ODE using
Maple
Use two different sets of coefficients Set 1
a2 and g2 Set 2 a2 and g0.0001
14
Maple commands and output
15
Maple commands and output
16
Maple commands and output
17
Famous ODE Simple pendulum
Second order non-linear ODE, because we have
second derivative.
To solve analytically, make the approximation
sinqq. This makes the ODE linear for small
amplitude oscillations.
18
Simple pendulum linearized ODE
This is called the harmonic oscillator equation.
19
Solve harmonic oscillator problem using Maple
20
Plot harmonic oscillator solution
21
Put 2nd order ODE in standard form as two 1st
order ODEs
22
Solve pendulum problem using standard form ODEs.
23
Solve pendulum problem using standard form ODEs
24
Plot pendulum solution functions
25
Systems of ODEs

• Systems of ODEs are solved using Maple in much
  the same way as a single ODE.
• Systems of ODEs are often needed when two or
  more changing functions depend on each other.

26
A predator-prey model
27
Seek equilibrium solutions for rabbit and fox
populations
28
Try rabbits and foxes
Yuck! Maple barfs on rabbits and foxes. Try
it.
29
(No Transcript)
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
Something isnt right here

• Weve found a set of two ODEs that Maple cannot
  solve analytically. To solve this problem we need
  numerical solution methods.