Higher-Order Differential Equations

So far we have considered differential equations

of the form

Such as

(separable equations)

(first-order linear equations)

Example

Write a differential equation for the path of the

mass and solve it.

Solution

Or if you prefer,

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

We found out that the guess will work, provided

Any linear combination of these two solutions is

a solution to the differential equation.

Example

Write a differential equation for the path of the

mass and solve it.

Solution

In fact, any linear combination of these two

solutions is again a solution. Thus, more

generally we have

In the last problem, we implicitly used several

facts from complex analysis.

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)

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)

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

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

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.

Inhomogeneous Equations with Constant Coefficients

Write the differential equation and solve.

Solution

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

Example

Solve the differential equation for a

mass-on-a-spring with sinusoidal external forcing

function.

Solution

Solution Analysis

In the long-term, the solution approaches a sine

wave.

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

Resonance

Example (Non-Resonant Sinusoidal Forcing)

Solution

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

Example (Resonant Sinusoidal Forcing)

Solution

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.

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.

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.

Weve considered three examples illustrating the

phenomenon of resonance

(No Resonance)

(Almost-Resonance, beats)

(Perfect Resonance)

Quantifying the Resonance Phenomenon

In the homework you will study this phenomenon in

a bit more generality.

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 ?

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

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,

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)

Additional Topics in Higher-Order Equations

Electric Circuits

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.

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.

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

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.

Summary

Variation of parameters works whenever we have an

equation of the form

(http//math.ucsb.edu/sideris/Math243-F09/notes-0

9-10.pdf )

Applications

blah blah blah

(No Transcript)

Applications

Blah blah blah

Higher-Order Differential Equations

So far we have considered differential equations

of the form

Such as

(separable equations)

(first-order linear equations)

Example

Write a differential equation for the path of the

mass and solve it.

Solution

Or if you prefer,

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

We found out that the guess will work, provided

Any linear combination of these two solutions is

a solution to the differential equation.

Example

Write a differential equation for the path of the

mass and solve it.

Solution

In fact, any linear combination of these two

solutions is again a solution. Thus, more

generally we have

In the last problem, we implicitly used several

facts from complex analysis.

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)

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)

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

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

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.

Inhomogeneous Equations with Constant Coefficients

Write the differential equation and solve.

Solution

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

Example

Solve the differential equation for a

mass-on-a-spring with sinusoidal external forcing

function.

Solution

Solution Analysis

In the long-term, the solution approaches a sine

wave.

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

Resonance

Example (Non-Resonant Sinusoidal Forcing)

Solution

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

Example (Resonant Sinusoidal Forcing)

Solution

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.

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.

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.

Weve considered three examples illustrating the

phenomenon of resonance

(No Resonance)

(Almost-Resonance, beats)

(Perfect Resonance)

Quantifying the Resonance Phenomenon

In the homework you will study this phenomenon in

a bit more generality.

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 ?

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

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,

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)

Additional Topics in Higher-Order Equations

Electric Circuits

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.

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.

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

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.

Summary

Variation of parameters works whenever we have an

equation of the form

(http//math.ucsb.edu/sideris/Math243-F09/notes-0

9-10.pdf )

Applications

blah blah blah

(No Transcript)

Applications

Blah blah blah