Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Various Problems

Hello.

I am practicing a variety of general problems and I was hoping I could get my solutions verified so I can see whether I am understanding the concepts or not. I am putting them all into this one post so not to flood the forum with individual posts. My solutions will be in the posts following.

Thanks in advance.

1. Solve the following:

a.

b.

c.

2. Plot the phase portrait of the system and answer a - c.

a. What are the equilibrium points of the system?
b. What is the long term behavior of the system with initial conditions x(0) = 3 and y(0) = 1?
c. Plot the x(t) and y(t) planes.

3. Find the equilibrium points:

4. Plot the phase portrait and do a - c.

a. Determine the long term behavior of the system.
b. Plot y(t).
c. Solve the differential equation.

5. Consider the partially decoupled system:

a. Derive the general solution.
b. Find the solution that satisfies the initial condition (x0,y0) = (-1,3)

6. Solve the differential equation:

7. Use Euler's Method for Systems to find y(1) given x(0) = 0, y(0) = 2, with a step size of .1 for the system

I'll be working these out throughout today and tomorrow. Thanks again!

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

For problem number 7 I am required to use a program written in class and executed in Maple. I get that y(1), when t equals 1, is 1.342224239.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

For problem 6, let

and

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

For problem 5

Solving the second equation yields

Plugging this into the first equation gives

Solving this for x(t)...

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

Me again. Solving problem 1,

a.

Answer:

b.

Answer:

c.

Answer:

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Various Problems

hi Reuel,

I've checked Q6.  That looks correct to me.

Q5.  I've stopped thinking about this one.  I think this is my best shot. 

Shouldn't it be

That gives me

Q1a  OK

Q1b OK

Q1c OK so far.

Q7.  I haven't got Maple.  Can I do this in Excel ?  Hhmm, not sure.  I've tried it and get y(1) = 1.844....  So maybe I cannot do it this way.

Over to you bobbym. 

Bob

Last edited by Bob (2011-02-23 03:52:59)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

#2....

a. Equilibrium points at x = y = 1

b. See attached documents.

c. I am not sure how to plot an individual plane?

Last edited by Reuel (2011-02-23 03:02:02)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Various Problems

hi

a. Equilibrium points at x = y = 1

Ok

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

Problem 5 worked out:

a. Derive the general solution.

How about that one? There may still be a mistake somewhere, I dunno.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

Problem 3:

Equilibrium points at

and

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Various Problems

Q1c OK so far.

What more is supposed to be done with this sort of problem? I am not sure how to solve for a second order differential equation. That is also what I am kind of wondering about #4.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Various Problems

hi

See my earlier post on Q5.  I used e^(-t) as an 'integrating factor'.  You multiply all terms by it.

The integral of (1 over e^t) = e^(-t), a few lines down doesn't need a 'ln' (natural log)  as it is directly integrable to -e^(-t).

My final x checked out ok when I differentiated it.

For 1c, cannot you do the same as in Q6 ?

Q3.  Correct.

Bob

Last edited by Bob (2011-02-23 05:21:33)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Various Problems

Over to you bobbym.

Hi Bob;

How are you?

Okay I am here, what now? Does anyone want to hear my Sylvester Stallone impersonation? I do a good Marlon Brando too. That always brings down the house. No one wants me for my real talent, impersonations. They could make a blooper tape out of my recent attempts at math.

For #7:

Using the forward Euler method ( what a name ) I am getting:

t = 1
x = 1.797
y = 1.3422

This is with a step size of .1.

Hi Reuel;

How are you? I would need to see your Maple code to comment about it. Also you do not need to write a routine to do this the Maple assistants do a really good job.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Various Problems

hi bobbym,

I'm great thanks.  Just finished decorating the lounge.  Got some family over and we're watching Mama Mia.  So I might just have to sing along too.  You can impersonate Piers Bronson, if you want.

So, about this Euler method.  You have Reuel's value, so I bow to your superior calculating.    I'm assuming you calculate dy/dt and dx/dt for the starting values of x and y and then take a short step of dt = 0.1 to get dx and dy (apologies for poor notation here) and then use these to step up y and x.  Then do this over until you've done it 10 times.

As I read this over I realised a mistake with my Excel formulas so I have just corrected them in a spirit of optimism.

Of darn it;  I've made it worse; now I've got y(1) = 1.984   

Later edit.

Oh joy, I've worked it out.  I had my angle in degrees not radians.  Now it works and I get the same as you. 

Inspired by ABBA, maybe ?

Bob

Last edited by Bob (2011-02-23 09:50:21)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Various Problems

Hi Bob;

I am glad you got it This is the algorithm for what I did.

If t ≠ 1 Then 1)

I have never seen Mama mia!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Various Problems

hi bobbym,

That's much like mine once it's translated into MS functions.  Except that where I should have had

I had converted x into radians, because I am so used to doing that for trig in Excel, forgetting that it was already in rads.

Oh well you learn something every day!

As for not having seen Mama Mia, I'm sure ABBA fans will forgive you.  I've never seen The Sound of Music!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Various Problems

I have never seen it either.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.