# Higher-Order Differential Equations - PowerPoint PPT Presentation

Title: 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
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
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
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## Higher-Order Differential Equations

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Title: 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
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
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