----------------------------------------------------------------------------- <19> Problem 10.2.* ----------------------------------------------------------------------------- Just experiment. The idea is to choose points from all regions (i.e. the eigenvectors divide the plane into four regions, choose a point in each) The IC are set in the plot command [[x(0),y(0)]=(0,1), [x(0),y(0)]=(1,0),.....] and you can choose as many as you want (but then you must define the color for each graph, i.e. you need to define as many colors as you have IC V. > I have noticed that in the maple script you have for plotting on this > assignment you have 5 initial conditions. I can't figure out how to choose > these initial conditions for #13 which is: > > dx/dt=x-6y > > dy/dt=2x-7y > > with roots r=-1,-5 > > which is improper and asymptotically stable. > > > > > in your example you write: > > "In this example, we are looking at x'=x+2y,y'=5x-2y, with five sets of > initial conditions: > > but then in the script you have: > > DEplot([diff(x(t),t)=2*x(t)+y(t),diff(y(t),t)=5*x(t)-2*y(t)], > > in other words, the script has x'=2x+y not x'=x+2y > > which one is correct and how did you find these initial conditions? ----------------------------------------------------------------------------- <18> Problem 4.12.5 ----------------------------------------------------------------------------- (a) use formula (8), p.241 with b=0, and solve for A (the coeff. of the homogeneous solution): (B and theta are known; A and phi are to be determined; tan(theta) = (k-m*gamma^2)/(b*gamma) -> infinity, so can take theta = (+/-)pi/2) i.e. the sign depends on whether theta is +pi/2 or -pi/2 ; that is + if gammaomega) since y=y'=0 at t=0 substitute to find A*sin(phi) + B*sin(theta) = 0--> A*sin(phi) = -B*sin(theta) =-(+/-)B A*omega*cos(phi) + B*gamma*cos(theta) = 0 --> A*omega*sin(phi) = 0 So conclude sin(phi) = 0 --> phi = 0 Now you can find A = -(+/-)B = (-/+)B and the formula becomes: y = A*cos(omega*t+phi) + B*cos(gamma*t + theta) = (-/+) B* cos(omega*t + pi/2) + B*cos(gamma*t+theta) (b) This needs the trig. identity: cos(a+b) = cos(a)cos(b)-sin(a)sin(b) (1a) cos(a-b) = cos(a)cos(b)+sin(a)sin(b) (1b) Adding: cos(a-b) - cos(a+b) = 2sin(a)sin(b) (2) To turn this around: cos(A) - cos(B) = 2 sin((A+B)/2)sin((B-A)/2) (3) (this follows from (2) by setting: A = a-b and B = a+b i.e. a = (A+B)/2, b = (A-B)/2) Here identify a = omega*t, b = gamma*t (the other sign works likewise) ----------------------------------------------------------------------------- ----------------------------------------------------------------------------- <17> Using maple to find answers to hw. problems ----------------------------------------------------------------------------- Here is a sample Maple session that gets you the general solution (set C1, C2 = 0 to find particular solution) of a nonhomogeneous DE with constant coefficients. You may use it to check your answers to the even homework problems (possible quiz material). >restart: with(DEtools): >de1 := D(D(y))(x) + 2*D(y)(x) + y(x) = t*exp(-t); >gsolution := desolve({de1},y(x)); ----------------------------------------------------------------------------- <16> Sec. 4.8 ----------------------------------------------------------------------------- > > For homework in section 4.8 do our particular equations have to include > > solving the values for A, B, C, etc? If so, it is my understanding from > > today's lecture that this is done in the following manner: > > > > find y, y', and y'' > > sum y, y' and y'' (Are these always summed, or does it depend on the > > given equation. i.e., y" -'y-2y) ******* it depends on the given equation. i.e., y" -'y-2y = 0 > > solve for A B C, etc > > > > Is there another way to do this? It seems very time-consuming and > > tedious. ******* there is a bag of algebraic tricks to make this type of calculation more streamlined. The theory of differential operators gives the necessary tools. The process is so automatic that it can be done by computer, using a Computer Algebra System like Maple. We will see a few tricks tomorrow (and I will make some Maple scripts that do the job available). For now it is good training to do it by hand; there is an understanding one gains of the whole process that is worth taking the algebra walk, at least to the extent of doing the homework (it is not that bad!) ----------------------------------------------------------------------------- <15> Homework mistakes (taking elasticity to new limits...) ----------------------------------------------------------------------------- > I also wanted to ask you if it was possible to turn in homework > assignment from sections(4.3 & 4.4) late? I misread the Homework > assignment I turned in section 4.2 today instead of (4.3 & 4.4). > I think a few other students accidently did the same thing, because I > saw 4.2 on a few other students homework as they were turning it in. > This mistake probably occured when we looked at the dotted lines that > seperate the weeks on the Homework Assignments web-page. I really > don't want to loose the points from that assignment. If you want I > could work some differnt problems from those sections if you have > already posted the solutions. > ********** (THIS ALSO APPLIES TO ANYONE ELSE WITH THE SAME EXCUSE!) OK! you can turn in the other problems (the ones w/o the stars) from the same sections that you missed. To get credit they ought to be in Thursday together with the regularly due homework. ----------------------------------------------------------------------------- <14> Problem 48 of section 4.8 pg.211 ----------------------------------------------------------------------------- > helped me complete the assignment in section (4.8). I did have one > more question for you: how do we solve the homogenious portion of a > higher-order differential equation? > For example: y''''-3y''-8y=0 > Problem 48 of section 4.8 pg.211 > I was able to solve for the particular solution, but I had to assume > that no terms of the homogenious solution would be a corresponding > solution to the particular solution. For the particular solution I > got: Yp = -(1/4)sinx > ********************************* It is easy to check if it is a homogeneous solution: just plug it in and see if it makes the equation = 0. On the other hand, this is exactly what you did anyway! The fact that you were able to solve the problem proves that what you plugged in was not a homogeneous solution... Now to answer your original question: in general you try the same idea, i.e. y=exp(r*t) and arrive at a polynomial for r. Here r^4-3*r^2-8=0 This is what is called a biquadratic equation and you can convert it to a quadratic by the substitution s=r^2 then s^2-3*s-8=0 -> s = (3 +/- sqrt(9-32))/2 = (3/2) +/- i*sqrt(23)/2 from this you find r = +/- sqrt{(3/2) +/- i*sqrt(23)/2} and this way we get 4 solutions (you still need to find the square root of that complex number, but I don't want to take all the fun away!) (I am also posting a short writeup on the solution of cubics) ----------------------------------------------------------------------------- <13> MISSING BAG: DID ANYONE SEE IT? PLEASE CONTACT ME! ----------------------------------------------------------------------------- > I was wondering if anyone found a small orange bag after class on > Tuesday. **** unfortunately I did not hear about a bag. There was only an (orange) canteen on the desk when I left the room, and there were already students from the next class so I would not necessarily notice any objects on the floor. Hope you find it. ----------------------------------------------------------------------------- <12> Maple 8 vs. Maple 7 (I believe!) ----------------------------------------------------------------------------- > I tried your maple commands with problem 7, but I ran into a little > trouble and could not contact you. On the command simplify(%/exp(r*t)); > the computer returned an error message that said that the / sign was > unexpected. Did I read the command wrong or is there something I can do > to get around this next time? ******* The Maple problem may be related to version differences. This command works on my own (and the Math. Dept.'s varsion 8) but I think in Maple7 it ought to read: simplify("/exp(r*t)); ******* Of course I am assuming that all previous commands were executed in correct order! ----------------------------------------------------------------------------- <11> Subject: Re: section 4.11 #5 2/11/03;23:55 ----------------------------------------------------------------------------- y"+10y'+ky=0 r^2+10r+k=0 r = -5+/-sqrt(25-k) This leads to the 3 cases as shown in the back. For the values given you get each case (2 real, 1 real, 2 complex) The only thing that is a bit "unusual" in the answers is the combination: A*exp(u*t)*cos(v*t)+B*exp(u*t)*sin(v*t) = exp(u*t)*[A*cos(v*t)+B*sin(v*t)] = R*exp(u*t)*cos(v*t-p), with p = arctan(B/A), R = sqrt(A^2+B^2) This second form comes from the trig. formula: cos(a-b)=cos(a)*cos(b) + sin(a)*sin(b) So, let B |_______ | /| | / | | R/ | cos p = A/R , sin p = B/R , R = sqrt(A^2+B^2) | / | | / | | / p | |/______|_____________ A then A*cos(u*t)+B*sin(v*t) = R * [cos p * cos(v*t) + sin p * sin(v*t)] = r * cos(v*t - p) We will discuss this form more soon, when we work with forced oscillations... (If that was not where you had problems, why not send me your form, so I can be a bit more helpful?) ----------------------------------------------------------------------------- <10> Subject: math website access problems 2/11/03;13:30 ----------------------------------------------------------------------------- We are currently experiencing a problem with access to the math.unm.edu website from the following locations: ESC pod (confirmed) Zimmerman (reported) CSEL (reported) CIRT Networking is (Not?)working on the problem. ----------------------------------------------------------------------------- <9> Subject: Help with #28 (section 4.6). 2/10/03;13:40 ----------------------------------------------------------------------------- > When b=5: ->general solution is y(x)=[c1*e^(-x)]+[c2*e^(-4*x)] > ->use y(0)=1 and y'(0)=0 to solve for c1 and c2 for a > final answer of (show work!) > y(x)=[(4/3)*e^(-x)]-[(1/3)*e^(-4*x)] > When b=4: ->general solution is > y(x)=[c1*e^(-2*x)]+[c2*x*e^(-2*x)], > ->solve for c1 using y(0)=1 in general solution > ->then use y'(0)=0 in the equation ********** y'(x)=[-2*c1*e^(-2*x)]+[-2*x*c2*e^(-2*x)+c2*e^(-2*x)] > and solve for c2 ********* > > When b=2: ->general solution is > y(x)=c1*e^(-x)*cos[sqrt(3)*x]+c2*e^(-x)*sin[sqrt(3)*x] Again, set y(0) = 1, y'(0) = 0 to get two equations which can be solved for c1 and c2 ******* careful! be sure you differentiate the products e^(-x)*cos[sqrt(3)*x], e^(-x)*sin[sqrt(3)*x] >>>> CORRECTLY ! <<<< ----------------------------------------------------------------------------- <8> Subject : Re: math316 prob#30 sect.(2.6) ----------------------------------------------------------------------------- > I was having problems > with figuring out the promblems in section 4.1. i followed example one > but got stuck on the last part where they solved for A and B, > specifically on how they set one equation equal to one and anther one > equal to 0. I was wondering if you could help me out on this and explain > where the =sinwt fits into the problem. ************* Look at the web page: new link to "Set 5 Help" for how to work this. I did not manage to squeeze an example in class, so here is problem 4.1.4 (almost) for free - just need to run the maple or matlab script to get the graph. Apply the discussion to problem 4.1.1 ----------------------------------------------------------------------------- <7> Subject : Re: math316 prob#30 sect.(2.6) ----------------------------------------------------------------------------- ----- Message Text ----- > I would like to know if I am attemping to solve the problem#30 in > section 2.6 page84 correctly. I read the example titled "Equations > with Linear Coefficients", but I keep getting something I do not know > how to integrate. > > (x+y-1)dx + (-x+y-5)dy = 0 > > dv/du = [1+(v/u)]/[(1-(v/u)] ..... correct > > dv/du = (1+z)/(1-z) ..... "" > > z + udz/du = (1+z)/(1-z) ..... "" > > [(1-z)/(1+z^2)]dz = du/u > this equation breaks into the two (integrable) expressions: / / | [1/(1+z^2)]dz = arctan z; |[z/(1+z^2)]dz = (1/2)log(1+z^2) / / ----------------------------------------------------------------------------- <6> Thu. 1-30-03: Difficulties accessing the class web page ----------------------------------------------------------------------------- We verified that there is a problem and reported it-hopefully it will be fixed soon; let me know if you are still having problems w. access. ----------------------------------------------------------------------------- <5> Wed. 1-29-03: Difficulties accessing the class web page ----------------------------------------------------------------------------- Two people report difficulties with the web page; when this happens please send me email right away so I can see a pattern of times/locations. This way I can get somebody to look at the problem and fix it-or figure out what you did wrong. I have had no problem accessing the web page from home or from the terminals in DSH at all hours. ----------------------------------------------------------------------------- <4> Wed. 1-29-03: running maple@home etc: ----------------------------------------------------------------------------- > First, is it possible to run the program from home without installing > the program on your home computer. For example, can I log into unm and > access the maple program? You can access maple at UNM from home, but unless you have a fast connection you would be using the text-based interface (maple rather than xmaple) Follows what I get when I run maple on my telnet window: (works similarly on aix.unm.edu): -------------- styx[1.17] %maple |\^/| Maple 7 (SUN SPARC SOLARIS) ._|\| |/|_. Copyright (c) 2001 by Waterloo Maple Inc. \ MAPLE / All rights reserved. Maple is a registered trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. bytes used=1000360, alloc=851812, time=0.34 Warning, the protected names norm and trace have been redefined and unprotected > > de1:=diff(v(t),t)=1-v(t)^3/8; d 3 de1 := -- v(t) = 1 - 1/8 v(t) dt %%%%%%%%%%%%%%%%%%% Second, if I cannot run maple from home do > you recommend purchasing the program from the unm bookstore ($135 > student price version 8). Clearly it is best to have things run locally. However, if $135 is a substantial chunk for your budget, the text version that you can easily access with a dial-up connection is great for developing code; then you can do the graphics on campus. As you will notice from the web page, I am not partial to Maple; it is good for some things, but so is Matlab (and matlab is a lot easier to learn and has more forgiving syntax). Maybe you want to play with both for a while until you decide which is best for you. For the class either is adequate as far as plots and numerical solutions are concerned. Maple is superior for symbolic computations, Matlab for matrix-related and other numerical computations. Both have great graphics. %%%%%%%%%%%%%%%%%%%%% When I typed in the maple command > de1:=diff(v(t),t)=1-v(t)^3/8; > the output showed the partial derivative of v(t) with respect to the > partial derivative of t. Is that correct or should the output show the > derivative of v(t) with respect to the derivative of t. The graph that > maple generated using partial derivatives does look correct. whether you get the partial or total derivative symbol is a function of whether you are running the stripped-down text version or the fancy graphics version (xmaple; the windows version is the same as xmaple). The d versus del is just notation; in a sense, all derivatives in a symbolic context are pertial, since the constants are treated as independent variables until you replace them by numerical values. ----------------------------------------------------------------------------- <3> Wednesday 1-29-03: Homework 3 ----------------------------------------------------------------------------- 3( 1/28) Techniques for solving 1st order ODE: separable equs. 2.1,2.2 <2,4,7,12*,18*,38*> Means: read 2.1 and 2.2 ----------------------------------------------------------------------------- <2> Wednesday 1-29-03: Some tips on printing ----------------------------------------------------------------------------- To print directly from Maple click on the "printer" button. (produces a printer file, suffix .mws). To create a LaTeX file (which you can further edit or simply convert to "Postscript" and print): File->Export As->LaTeX....<> (give some filename with suffix ".tex", say abc.tex) Once this is saved in your directory, use a Unix window (i.e. not inside xmaple) and type: latex abc.tex; dvips abc.dvi -o abc.ps (the first command processes the typeseting script abc.tex produced by maple into the viewable file abc.dvi (view with "xdvi abc.dvi"); the second links it to the image files (suffix .eps) produced by maple for each plot and produces a postscript file abc.ps (viewable with: ghostview abc.ps). You now have a postscript file, abc.ps, which you can print. (You can easily incorporate this LaTeX file into larger documents, reports etc. Most scientific work is typeset in LaTeX, it will do you some good to get exposed to it early!) ----------------------------------------------------------------------------- ----------------------------------------------------------------------------- <1> Monday 1-27-03: MAPLE plots (question by Pete G.): ----------------------------------------------------------------------------- *In reaction to what you said thursday in class, all plots/graphs should be *done by computer, i am trying to convince maple to produce a direction *field plot for me. I am giving it the following command: *dfieldplot(diff(v(t),t)=1-(v^3)/8,v(t),t=0..6, v=-4..4, title=`section *1.3, problem 4`); *the result is a nicly typeset regurgitation of what i typed in. *How do i make it show me a graph? ==========> precede by invoking the DEtools package: >with(DEtools) >dfieldplot(diff(v(t),t)=1-(v^3)/8,v(t),t=0..6, v=-4..4, title=`section 1.3, problem 4`); or, follow the example on the web page (which also plots several solution curves): > with(DEtools): > de1 := diff(y(t),t) = y(t)*(y(t)-a)*(b-y(t)): > a := 1; b := 2: > DEplot(de1,y(t),t=0..4,{[0,3],[0,1.5],[0,.5],[0,.9],[0,-.1]},title='de1', axesfont=[TIMES,BOLD,10],titlefont=[TIMES,BOLD,14], labelfont=[HELVETICA,BOLD,12], arrows=THIN,linecolor=blue,y=-1..4); Either will produce the direction field ** i dont expect you to reply in time for me to produce plots for the *homework due thursday(i would need a reply 5 minutes ago), so i am going ----> TUESDAY!! These problems are due Tuesday (tomorrow) *to hand draw these graphs. I hope this is acceptable. ----> hand-drawn graphs are OK this time, but you need to learn how to compute them! -----------------------------------------------------------------------------