Spring 2009

# assignments

due Tue Jan 27

## getting started

• First: calculus review and Mathematica practice.
• See how far you can get with these, using pencil and paper, Mathematica, or another numerical tool of your choice.
• calc review
• Second: read chapter 1 in the text.
due Tue Feb 3

## numerical calculus

• Plot the derivative of e − 5x2 using at least two different methods.
• Do problem 21 on page 8: "A pond has initially on million gallons of water and and unknown amount of a nasty chemical. Water containing 0.01 gram per gallon of the chemical flows in at 300 gallons/hour. The mixture flows out at the same rate, so that the amount in the pond is constant. Assume that the chemical is uniformly distributed in the pond.
• Write down a differential equation for the amount of chemical in the pond at time t.
• How much chemical is in the pond after a long time goes by? Does that depend on the initial amount?
• Do problem 8 on page 17: A population p of field mice grows at a rate proportional to the population : dp / dt = rp .
• Find the rate constant r if the population doubles in 30 days.
• Find r if the population doubles in N days.
• Do problems 21, 22, 26 on page 25. (in edition 9).
due Tue Feb 10

due Tue Feb 17

## euler and logistic

• Finish at least one of the problems from last week that you haven't done yet, either the friction velocity squared one or the population growth one.
• Do these problems. (Here's a PDF.)
due Thu Feb 19

due Tue Feb 24

## forced damped oscillator

• Start reading chapter 3, particularly 3.1 (intro), 3.3 (complex roots), 3.7 (oscillator), 3.8 (forced).
• Finish what we started in class, namely :
1. Given a simple harmonic oscillator my'' = − ky with a given initial conditions y(0) = y0,y'(0) = v0 , write down the solution and verify that it work.
2. Verify that the damped oscillator can be solved by equations of either the following forms :
• y(t) = AeBtsin(Ct + D)
• y(t) = GeHt where both G and H may be complex.
• Explain how the real (A, B, C, D) are connected to the complex (G, H).
due Thu Mar 5

## resonance

• Make sure you can follow my Mathematica notes for Feb 24 and Feb 26, and read the corresponding parts in chapter 3.
• Find u(t) where 3u'' − u' + 2u = 0 with u(0) = 2,u'(0) = 0 . For t>0, find the first time where | u(t) | = 10 . (Problem 23, page 163.)
• Show that the two solutions of (i) the over damped oscillator, and (ii) the under damped oscillator are independent, by calculating their Wronskian.
• The equation for the RLC circuit for (t,I(t)) where I is the current, is L(d2I) / dt2 + RdI / dt + I / C = 0 . (See for example pg 201 in the text.) Put this equation into the same dimensionless form as in my oscillator2 notes, and describe what the dimensionless constants are in terms of the resistance R, capacitance C, and inductance L.
• A vibrating system obeys the equation u'' + au' + u = 0 . Find the value of the damping term "a" for which the "period" of the damped motion is 50% greate than the period of the undamped motion. (problem 13, page 203).
• A mass weighing 16 punds stretches a spring by 3 inches. The mass is attached to a damper with a constant of 2 pounds/(ft/sec). The mass is set in motion from its equilibrium position with a downward velocity of 3 inches/sec. Find its position at any later time t, and find the time time tau such that |u(t)|< 0.01 inches for t>tau. (problem 10, pg 203)
due Fri Mar 6

• a place for Jim to record midterm grades
due Thu Mar 12

## numerical practice

• Read sections 8.1 and 8.2 .
• Write (in Mathematica or Python) a numerical program to implement the "improved Euler" algorithm described in the text, for solving 1st order equations in the form dy/dx = f(y,x) starting at some (x0, y0).
• Find an explicit estimate of the error in one step of this algorithm, using midpoint values of (x,y) or dy/dx, or values averaged from the endpoints of the interval found with Euler, Backward Euler, or the Improved Euler method.
• Use the results of the previous estimate to iteratively improve your solution by automatically picking a smaller step until some desired fractional error is reached.
• Explore the solutions to dy/dx=sin(x)/x near x=1 using these ideas, and come to class ready to show how far you got.
due Tue Apr 7

## runge kutta etc

• Review Euler's method, Heun's algorithm, and programming these in Mathematica.
• Read about Runge-Kutta in the text, chap 8.3, or other online source.
• Do problem 14 in the text, page 463 :
• dy/dt = t2 + y2
• initial value t0 = 0, y0 = 1
• draw a direction field
• Implement RK4 and use it to find approximate values for the solution at t=0.8, 0.9, 0.95. Choose a step size small enough that your results are accurate to four digits.
• Graph the solution.
• Do the same calculation with another algorithm such as adaptive step or Heun; compare the accuracy, step size, and dy/dt evaluations.
• What happens at t=1?
• Read section 8.6, on systems of equations. A 2nd order equation in d2y/dt2 and dy/dt can be reduced to a system two 1st order equations in dy/dt and dv/dt , where v=dy/dt.
• Do problem 8 on pg 43:
• Consider the initial value problem x'' + t2 x' + 3x = t, with x(0)=1, x'(0)=2.
• Convert this problem to a system of two first order equations.
• Find and graph solutions to t=0.5 and t=1.0 using runge kutta. (Suggestion: h=0.1)
due Tue Apr 14

## eigens

• Read chapter 7 in the text.
• Describe just what this "eigen" stuff is all about.
• Find the egienvalues and eigenvectors of these matrices :
• A = {{1, 1}, {1, 0}}
• B = {{1, 1}, {4, 1}}
• C = {{4,1,1},{2,4,2},{1,1,4}}
• D = {{0, 1}, {-1, 0}}
• Set up the equations for 3 equal masses in a row connected by springs, like I did in class but with one more mass.
• What is the matrix?
• Find the three "normal modes" and frequencies of oscillation of this system, and explain in words how the masses are moving in each case.
due Tue Apr 21

## test 2

• pdf or notebook
• (Note that the assignment below this is also due on the same date.)
due Tue Apr 21

## systems of equations

• Read either/or chapter chapter 7, http://openlearn.open.ac.uk/file.php/2747/MST209_7.pdf, my class notes, or other description of using eigens to solve systems of 1st order linear differential equations.
• Proceeding as I did in class on Thurs (using the workbook from Tues notes), solve this system of equations :
• dx/dt = y
• dy/dt = - 4 x
• with the initial conditions y(0) = 1 ; x(0) = -0.25 .
• And as I did in class, find two particular solutions, one general solution, and sketch the phase portrait.
due Tue Apr 28

## stability and critical points

• Read 9.1 and 9.2 in the text.
• and/or start in on egwald's systems of equations notes
• Do problems 1, 5, 16 , section 9.1
• Do problem 8, section 9.2
• For problem 8, generate a composite plot with a direction field and a few numerically calculated trajectories.
due Tue May 5

## chaos

• Explore at least two of the "Wolfram demos" that I pointed to in the Apr 30 notes, which connect to work that we did this semester.
• For each of them, give a link to the demo, and provide a brief explanation of what's going on and how the ideas we explored this semester connect.
due Fri May 8