Higher-Order Differential Equations

1
Higher-Order Differential Equations
2
So far we have considered differential equations
of the form
 
Such as
 
(separable equations)
 
(first-order linear equations)
 
 
3
Example
 
 
Write a differential equation for the path of the
mass and solve it.
Solution
 
 
 
Or if you prefer,
 
4
For the mass-on-a-spring problem, we got the
second order differential equation
 
More specifically, this is called a second order
homogeneous ordinary differential equation with
constant coefficients
 
All such equations are solved in the same
straight-forward way
 
 
The equation in this case becomes
 
 
 
5
 
We found out that the guess will work, provided
 
 
 
 
 
Any linear combination of these two solutions is
a solution to the differential equation.
 
 
 
 
 
6
 
7
Example
 
 
Write a differential equation for the path of the
mass and solve it.
Solution
 
 
 
 
8
 
 
 
 
 
 
 
 
 
 
 
 
 
In fact, any linear combination of these two
solutions is again a solution. Thus, more
generally we have
 
 
 
 
9
 
10
In the last problem, we implicitly used several
facts from complex analysis.
 
 
 
 
11
In the last two problems, we had differential
equations of the form
 
 
 
 
In general, this same technique works for
homogeneous equations of any order with constant
coefficients.
Just guess
To solve
 
 
And solve the characteristic equation
 
 
The general solution is then
(see repeated root exception. Interpret complex
roots with Eulers identity)
 
12
Example
Write the general solution to the differential
equation
 
Solution
We get a characteristic equation
 
 
(if you are interested in factoring algorithms,
consult a course or text on Abstract Algebra)
 
 
13
Example (Repeated Roots)
Write the general solution to the differential
equation
 
Solution
As usual we write and solve the characteristic
equation
 
 
 
 
 
Actually the correct solution here is
 
 
14
Example
Write the general solution to the differential
equation
 
We solve the characteristic equation
Solution
 
By factoring
 
 
Using the repeated root rule, the generals
solution is
 
15
Im not aware of applications of these types of
equations except for 2nd order equations.
Bonus points for finding and sending me an
interesting application leading to a 3rd- or
higher-order ODE with constant coefficients.
 
16
Inhomogeneous Equations with Constant Coefficients
17
 
 
 
Write the differential equation and solve.
Solution
 
 
 
 
 
 
 
 
 
 
18
 
 
 
Write the differential equation and solve.
Solution
 
 
 
To find all solutions, we first solve the
complimentary homogeneous equation obtained by
replacing the right-hand-side with zero.
 
 
 
 
 
Now we solve the initial conditions
 
19
 
20
Example
Solve the differential equation for a
mass-on-a-spring with sinusoidal external forcing
function.
 
Solution
 
 
 
 
 
 
 
 
 
 
 
 
21
 
22
Solution Analysis
 
 
 
In the long-term, the solution approaches a sine
wave.
23
Review of analytic trigonometry
 
It is helpful to rewrite these using the identity
 
 
 
 
This is more useful because it tells us the
amplitude and phase displacement of the
oscillation.
I suggest you read the documentation on the very
helpful two-argument ArcTan function
24
Resonance
25
Example (Non-Resonant Sinusoidal Forcing)
 
 
Solution
 
 
 
 
 
 
 
 
 
 
26
Example (Resonant Sinusoidal Forcing)
 
 
Solution
 
 
 
However, when we plug it into the differential
equation we get
 
 
 
 
Plugging this into the differential equation we
now get
 
 
 
 
 
 
27
Example (Resonant Sinusoidal Forcing)
 
 
 
Solution
 
 
 
 
 
 
28
The last two problems deserve a comparison.
 
 
 
 
The phenomenon on the right is called
resonance It occurs when the forcing frequency
matches the natural oscillating frequency of the
system.
29
Example (Almost-Resonance)
Solve the differential equation
 
Solution
I leave computation details as an exercise. In
the end we get
 
We can shed light on this behavior through the
use of trigonometric identities.
30
 
 
Recall
 
 
 
 
 
Therefore,
 
The first term is much larger than the second
term, and explains the behavior of the solution.
 
This phenomenon exhibited here is called beats,
and occurs in the theory of sound vibration.
31
Weve considered three examples illustrating the
phenomenon of resonance
 
 
 
(No Resonance)
(Almost-Resonance, beats)
(Perfect Resonance)
32
Quantifying the Resonance Phenomenon
 
 
 
 
 
 
 
In the homework you will study this phenomenon in
a bit more generality.
33
Example (Resonance with Resistance)
Find a particular solution to the mass-spring
system with resistance and sinusoidal force
 
Solution
 
The characteristic equation is
 
 
 
We obtain the system of equations
 
Again we solve by computer ?
34
Example (Resonance with Resistance)
Find a particular solution to the mass-spring
system with resistance and sinusoidal force
 
Solution
Our solution is therefore
 
where
 
Our solution can be rewritten
 
 
Note that we have a nice expression for the
amplitude
 
 
35
Optimal Resonating Amplitude
Given the differential equation
 
We found a formula for amplitude of the solution
 
 
In the homework, I ask you to show that the
optimal resonating frequency and amplitude are,
respectively,
 
 
 
 
36
Example While driving on a bumpy road, a certain
car bounces up-and-down on its shock absorbers.
 
 
Solution
 
 
 
 
(four 20cm up-and-down jerks per second)
37
Additional Topics in Higher-Order Equations
38
Electric Circuits
39
 
 
 
 
40
We will not discuss this equation since you
already know how to solve itits the same
equation as for the mass on a spring with
resistance
 
 
It is often the case, in the sciences, to come
across dual systemsseemingly-different phenomena
represented by the same differential equations.
41
Variation of Parameters
Well do one example using an old (18th century)
but powerful technique, variation of parameters.
Later, when we study systems of differential
equations, well study a generalization of this
technique that works in an even broader variety
of situations, has theoretical importance, and
works great with computers.
42
Example
Find a particular solution to the differential
equation
 
Solution
 
 
 
 
 
 
Take a derivative
 
 
 
Take another derivative
 
Plug this into the differential equation to find
that
 
 
 
43
Example
Find a particular solution to the differential
equation
 
Solution
 
This will be a solution provided we can solve the
equations
 
 
 
This simplifies down to
 
Integrate
 
 
 
Evidently, guessing such a solution is out of
the question.
44
Summary
Variation of parameters works whenever we have an
equation of the form
 
 
 
 
 
45
(http//math.ucsb.edu/sideris/Math243-F09/notes-0
9-10.pdf )
Applications
blah blah blah
46
(No Transcript)
47
Applications
Blah blah blah