Linear Differential Equations

Agnishom · 2015-04-02 14:02:43

How do I solve the following system:

v' = 4v - 5w
w' = 2v - 3w

bobbym · 2015-04-02 17:16:41

Hi;

Start by writing the whole thing in matrix form:

The eigenvalues of

are {2,-1} and the eigenvectors are

The solution to that set of equations is then

Agnishom · 2015-04-02 22:01:32

What are Eigenvalues?

Bob · 2015-04-02 22:28:35

hi Agnishom,

In the vector equation:

A is a square matrix of size equal to the dimension of the vector b, and lambda is a scalar.

b is called an eigenvector of A and lambda is its associated eigenvalue.

In the two by two case, if you give the vector, components of u and v and write out the simultaneous equations that come from the above, you can find b and lambda. It is not unusual for a 2 x 2 matrix to have two of these.

http://en.wikipedia.org/wiki/Eigenvalue … genvectors

Bob

In bobbym's post he has switched from v and w, to u and v respectively, but it comes out the same. I'll forgive him as I expect he is still a bit shaken after the alligator incident.

Last edited by Bob (2015-04-02 22:37:28)

Agnishom · 2015-04-02 23:16:01

How does one find the eigenvalues and then solve the rest of the problem?

Last edited by Agnishom (2015-04-02 23:23:31)

Bob · 2015-04-03 00:05:42

Like this:

Do the matrix multiplication to get

Now there are three unknowns and only two equations but that often doesn't matter with vectors because if u is a solution so is ku for any scalar k. So you only need to find a simple vector from an infinite set to get your solution.

Make lambda the subject of each and equate to eliminate lambda:

So we can choose

And you can go back and substitute these values to get the two values for lambda.

Bob

Agnishom · 2015-04-03 00:30:29

Brilliant.

Can you explain the next step?

Bob · 2015-04-03 01:07:33

Partly.

When something differentiates to give something like itself, it's usually an exponential. (There are, no doubt, learned works explaining why this is so but I don't have this in my brain. But lots of integration consists of using a trick to get a possible answer and then observing that it is indeed the answer. If it works you might be content with that. For my own satisfaction, with a recent question where an exponential expression provided a solution I did rough out a proof that any other solution (however obtained) would have to be a scalar multiple of the exponential. I could probably do the same here if you're really keen to see the full pure mathematical solution. Or you could just wait for Olinguito to teach us both.

So using, for example,

a solution is:

proof:

Similarly the other particular solution. Then construct a linear combination of these two with arbitrary constant c1 and c2 and there you are.

That's the extent of my knowledge on this.

Bob

zetafunc · 2015-04-03 02:19:52

This is really just an application of diagonalisation. Let's consider the following system:

Suppose that

and that

.

Then, if ' denotes differentation with respect to t, we write x' = Ax. We now make a change of variable. Put x = Py, where P is invertible, and:

and

Then, differentiating with respect to t, x' = Py' so that the system x' = Ax reduces to Py' = APy. That is:

Py' = APy => y' = P⁻¹APy.

Does P⁻¹AP look familiar to you?

Last edited by zetafunc (2015-04-03 02:25:44)

Agnishom · 2015-04-03 04:03:24

No, what is it supposed to be?

Bob · 2015-04-03 04:08:16

hi zetafunc

Great to hear from you again. How are the studies going?

It's been a loooong time since I saw P⁻¹AP.

How do you know it is going to be diagonalisable?

For Agnishom:

This might help:

http://en.wikipedia.org/wiki/Diagonalizable_matrix

Bob

Olinguito · 2015-04-03 04:27:09

Let

[list=*]
[*]

[/*]
[/list]

and suppose

[list=*]
[*]

[/*]
[/list]

for some v ≠ 0. Then

[list=*]
[*]

[/*]
[/list]

Since the LHS is zero for some x, y not both zero, the determinant of the matrix on the LHS must be 0.

Note that eigenvalues can be complex.

Bob · 2015-04-03 04:32:40

Thanks for that.

Do you know why we have to choose e^(something) for these, why no other solutions are possible

and also why the general solution is c1.(a particular solution) + c2.(another particular solution)?

Bob

bobbym · 2015-04-03 08:50:30

Hi;

I'll forgive him as I expect he is still a bit shaken after the alligator incident.

As most of you may have surmised my conflict with a gator was a fictitious event. In reality, there is no alligator with the intestinal fortitude to stand up to me. In the future, that may change...

How does one find the eigenvalues and then solve the rest of the problem?

The eigenvalue problem is considered one of the central problems of numerical analysis along with root finding and simultaneous linear equations. I have always wondered how there can be 3 centers?!

Boils down to solving for roots of a characteristic or auxiliary polynomial. When the systems are bigger, hand methods will breakdown. Take note of the story of how the great Alan Turing took a full month to get the roots of a 7th degree poly. Use this when the going gets tough and just keep motoring along...

Eigenvalues[{{4, -5}, {2, -3}}]

Agnishom · 2015-04-03 12:48:35

Eigenvalues are okay, what about the next part?

bobbym · 2015-04-03 16:15:57

Eigenvectors[{{4, -5}, {2, -3}}]

Bob · 2015-04-03 19:59:49

bobbym wrote:

As most of you may have surmised my conflict with a gator was a fictitious event. In reality, there is no alligator with the intestinal fortitude to stand up to me. In the future, that may change...

What! You made that up

Why will it change? Are you getting tastier? Or are alligators evolving to become less picky?

Bob

Agnishom · 2015-04-04 02:24:38

bobbym wrote:

Eigenvectors[{{4, -5}, {2, -3}}]

No, not that.

Can you explain what I can do when I already have these

Olinguito · 2015-04-04 02:27:36

bob bundy wrote:

Do you know why we have to choose e^(something) for these, why no other solutions are possible
and also why the general solution is c1.(a particular solution) + c2.(another particular solution)?

Last edited by Olinguito (2015-04-04 02:29:02)

Bob · 2015-04-04 03:19:42

Thank you Olinguito

Bob

Agnishom · 2015-04-04 03:39:37

Thank you bob, bobbym, Olinguito

bobbym · 2015-04-04 08:50:35

Hi;

Why will it change? Are you getting tastier? Or are alligators evolving to become less picky?

In the past alligators were afraid of man but due to silly laws they are no longer.

Agnishom wrote:

Can you explain what I can do when I already have these

The secret to effective problem solving is to learn by example. Most math types are trained to think theoretically, there are few if any examples in their books. Computer programmers are trained to think algorithmically and are more easily able to reason by following an example. Consider that when I provide an example it is a template or model for how to do all the rest you find. In this case all the rest of the types that have two distinct real eigenvalues.

pappym wrote:

Practice makes perfect.

Want to do another one and did you verify the solutions you already obtained?

Agnishom · 2015-04-04 13:32:20

I did verify them.

bobbym · 2015-04-04 17:31:15

Do you want to do another then?

Agnishom · 2015-04-05 02:11:40

Yes

Math Is Fun Forum

#1 2015-04-02 14:02:43

Linear Differential Equations

#2 2015-04-02 17:16:41

Re: Linear Differential Equations

#3 2015-04-02 22:01:32

Re: Linear Differential Equations

#4 2015-04-02 22:28:35

Re: Linear Differential Equations

#5 2015-04-02 23:16:01

Re: Linear Differential Equations

#6 2015-04-03 00:05:42

Re: Linear Differential Equations

#7 2015-04-03 00:30:29

Re: Linear Differential Equations

#8 2015-04-03 01:07:33

Re: Linear Differential Equations

#9 2015-04-03 02:19:52

Re: Linear Differential Equations

#10 2015-04-03 04:03:24

Re: Linear Differential Equations

#11 2015-04-03 04:08:16

Re: Linear Differential Equations

#12 2015-04-03 04:27:09

Re: Linear Differential Equations

#13 2015-04-03 04:32:40

Re: Linear Differential Equations

#14 2015-04-03 08:50:30

Re: Linear Differential Equations

#15 2015-04-03 12:48:35

Re: Linear Differential Equations

#16 2015-04-03 16:15:57

Re: Linear Differential Equations

#17 2015-04-03 19:59:49

Re: Linear Differential Equations

#18 2015-04-04 02:24:38

Re: Linear Differential Equations

#19 2015-04-04 02:27:36

Re: Linear Differential Equations

#20 2015-04-04 03:19:42

Re: Linear Differential Equations

#21 2015-04-04 03:39:37

Re: Linear Differential Equations

#22 2015-04-04 08:50:35

Re: Linear Differential Equations

#23 2015-04-04 13:32:20

Re: Linear Differential Equations

#24 2015-04-04 17:31:15

Re: Linear Differential Equations

#25 2015-04-05 02:11:40

Re: Linear Differential Equations

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