# Math 235, Spring 2015

### Announcements

 29 December Office Hours this semester. Mondays: 14:00-15:30 Wednesdays: 14:00-15:30 29 December Course materials: Required: Course packet Recommended: Differential Equations, 3rd edition by Blanchard, Devaney, Hall Both are available at the bookstore. 8 January The Festival of Scholars Coordinating Committee makes the following announcement: The Festival of Scholars is a campus-wide celebration of student work. It is an opportunity to discuss research, to exhibit, perform, or appreciate art, and to cross disciplinary boundaries. The Festival will be held on Friday, April 17, 2015. Classes will be cancelled on that day, but you are still required to participate in the Festival, either by presenting your work or attending presentations by your fellow students. 21 January Homework Day 1: Exercise 1.1 due Friday 23 January Special procedure: Submit to folder outside Paul’s office door. 21 January Homework Day 2: Exercises 2.1, 2.4, 2.6 due Monday 26 January. Submit in SQRC 24 January Typo Exam 1 is on Friday 13 February, the 11th day of class. Please correct page 14 of the packet. 26 January Class meets in Dubach Mac Lab today: If you have a Laptop, please install LaTeX ahead of time. See these resources. Mathematica: Tutorial 1, Tutorial 2 A LaTeX template for reports can be found here. 29 January Optional help sessions, Thursdays 12:30-1:30 in Howard 122 30 January Exercises 4.4, 4.6, 4.9 due in SQRC. Be sure to look at Exercise 4.7 2 February Exercises 5.1, 5.4, 5.5, 5.10, 5.12 due in SQRC 2 February Please work through the LaTeX tutorial before class. 2 February Some Mathematica code to complement today’s lecture: For the basic growth model $\frac{dP}{dt} = 0.5 P$: VectorPlot[{1, .5 P}, {t, 0, 5}, {P, -1, 1}] StreamPlot[{1, .5 P}, {t, 0, 5}, {P, -1, 1}]  Other examples: StreamPlot[{1, .5 P (1 - P/10)}, {t, 0, 5}, {P, -2, 12}] StreamPlot[{1, 1 - y^2}, {t, 0, 5}, {y, -2, 2}] StreamPlot[{1, y (1 - y) Sin[t]}, {t, 0, 5}, {y, -2, 5}]  2 February Announcements: Exam 1 is 13 February, during class time. Report 1 is due 16 February 4 February Exercises 6.1, 6.3, 6.5 due in SQRC 4 February Some Mathematica code relevant to today’s lecture: Plot the path $x(t) = e^t, y(t)= e^{2t}$ for $-1\leq t \leq 1$ ParametricPlot[{Exp[t], Exp[2 t]}, {t, -1, 1}, PlotRange -> {0, 10}]  Plot the vector field $\langle x+y,-x \rangle$ VectorPlot[{x + y, -x}, {x, -3, 3}, {y, -3, 3}] StreamPlot[{x + y, -x}, {x, -3, 3}, {y, -3, 3}]  6 February Exercises 7.3, 7.5, 7.6, 7.8, 7.9 due in SQRC Remember, SQRC closes early on Fridays 6 February Here is some code for today’s class $\frac{dy}{dt} = e^y$ with $y(0)=1$ and $\Delta t=.25$: deltat = .05; f[y_] := Exp[y]; values = RecurrenceTable[{ t[k + 1] == t[k] + deltat, y[k + 1] == y[k] + deltat*f[y[k]], t[0] == 0, y[0] == 1 }, {t, y}, {k, 0, 10}]; Grid[values] ListLinePlot[values, PlotMarkers -> Automatic]  $\frac{dy}{dt} = y^2$ with $y(0)=1$. We know solution is $y(t) = \frac{1}{1-t}$ deltat = .05; n = 15; f[y_] := y^2; soln[t_] := 1/(1 - t); values = RecurrenceTable[{ t[k + 1] == t[k] + deltat, y[k + 1] == y[k] + deltat*f[y[k]], t[0] == 0, y[0] == 1 }, {t, y}, {k, 0, n}]; Grid[values]; eulerplot = ListLinePlot[values, PlotMarkers -> Automatic]; actualplot = ListLinePlot[Table[{ values[[i, 1]], soln[values[[i, 1]]] }, {i, 1, n + 1}], PlotStyle -> Purple]; errorplot = ListLinePlot[Table[{ values[[i, 1]], Abs[values[[i, 2]] - soln[values[[i, 1]]]] }, {i, 1, n}], PlotStyle -> Red]; Show[eulerplot, actualplot] Show[eulerplot, actualplot, errorplot]  See also this page. 9 February Exercises 8.1, 8.3, 8.4 due in SQRC 9 February Here is a link to code for today’s lecture. 9 February Reminder: I am not collecting homework for Days 9 and 10. However, you are still responsible for learning the material. 11 February Code for today’s lecture. Day 12 Exercises 12.3, 12.5 due 18 February Announcement: Report 2 due 20 March Day 13 Exercises 13.1, 13.2, 13.3, 13.4 due 20 February Day 14 Exercises 14.2, 14.4 due 23 February Some comments about the first exam problem. Day 15 Exercises 15.3, 15.4, 15.5 due Wednesday 25 February Day 16 Exercises 16.1, 16.2 due Friday 27 February Sunday 1 March Day 17 Exercises 17.3, 17.4, 17.6 due Monday 2 March Day 18 Exercises 18.2, 18.3, 18.6 due Wednesday 4 March Day 19 Exercises 19.1, 19.2 due Friday 6 March Sunday 8 March Day 20 Exercises 20.1, 20.2 due Monday 9 March 8 March Comments about Report 1 Day 21 (9 March) Complete Exercises 21.1 and 21.2 today. You do not need to submit them. 9 March Selected solutions to Day 19 Exercises and Day 20 Exercises. Day 22 (March 11) Exercise 22.5, parts (3),(4),(5) due Friday 13 March (Sunday 15 March is OK) Day 23 (March 13) Exercises 23.2, 23.3, 23.4 23.5 due Monday 16 March Day 24 (March 16) Exercises 24.1, 24.3, 24.6 due Wednesday 18 March Day 25 (March 18) Exercise 25.1 due Friday 20 March March 18 Some tips for using Manipulate in Mathematica Day 26 (March 20) No homework! Day 27 (March 30) Exercises 27.1, 27.3 due Wednesday Report: Gravitation, due 20 April Day 28 (April 1) Exercises 28.1, 28.2, 28.3 due Friday Day 29 April 3 Exercises 29.1 and 29.2 due Monday I encourage you to think about Exercise 29.3 TYPO ALERT There is a typo in the formula for the gravitational report. Equation 26.1 should read: $\displaystyle\frac{d^2r}{dt^2} = r\left( \frac{d\theta}{dt}\right)^2 - \frac{G M}{r^2}$ Day 30 April 6 Exercises 30.2, 30.3 due Wednesday Day 31 April 8 Code for today’s class: Manipulate[Show[ Plot[.5 x^2, {x, -4, 4}, PlotRange -> {0, 4}, AspectRatio -> Automatic], StreamPlot[{Sqrt[2 H - x^2], -b (2 H - x^2)}, {x, -4, 4}, {H, 0, 4}]], {b, 0, 3}]  and Manipulate[Show[ Plot[-Cos[x], {x, -5, 5}, PlotRange -> {-2, 3}, AspectRatio -> Automatic], StreamPlot[{Sqrt[2 H + 2 Cos[x]], -b (2 H + 2 Cos[x])}, {x, -5, 5}, {H, -2, 3}]], {b, 0, 1}]  Day 31 April 8 Exercises 31.1, 31.3 due Friday Day 32 April 10 Problems 1(c), 1(d) 1(e), 4, 5 due Monday Updated April 13 to correct typo. (Assignment was correct as given in class.) Day 33 April 13 Problems (2) and (3) due Wednesday. Day 34 April 15 Exercises 34.2, 34.3, 34.7 due Friday Sunday. Due date extended due to SQRC closure (Festival of Scholars). Day 35 April 17 Festival of Scholars — no homework. I encourage you to be working on your report “Gravitation”! Day 36 April 20 Exercises 36.1, 36.4 due Wednesday Day 37 April 22 Exercises 37.2, 37.4, 37.5 due Friday