Week 4: 14 Sep, |
15 Sep, | 16 Sep, | 17 Sep, | 18 Sep, |

The Jules Verne problem [slides]. Finished last week in 12:25, finished today in 7:30. Partial fractions. Method of sampling [clear fractions, substitute samples, solve for A,B, ...] Method of atoms [clear fractions, multiply out and match powers, solve for A,?B,...] Heaviside's cover-up method [partially clear fraction, substitute root, report constant] Separation of variable solutions with partial fractions. Exercise solutions to the four problems due in 2.1, 2.2. Phase line diagrams. Phase diagram. Postponed: Verhulst models with harvesting term. Differences in phase diagrams of non-autonomous systems. What new things we see in computer graphics of such models.

- The tennis ball problem. Does it take longer to rise or longer to fall?

- Jules Verne problem. A rocket from the earth to the moon.

Problem notes for 2.3-10. 2.3-20 (1.8 K, txt, 24 Jan 2010)

Lecture on symbolic solution exam review problems ER-1, ER-2 [identical to maple L3.1, L4.1], which are due next week. Done in 12:25 last week.

Introduction to Rect, Trap, Simp rules from calculus.

Introduction to the Euler, Heun, RK4 rules from this course.

Dot tables, connect the dots graphic.

- How to draw a graphic without knowing the solution equation for y.
- Key example y'=sqrt(x)exp(x^2), y(0)=2.

Challenge: Can you integrate sqrt(x) exp(x^2)? - Making the dot table by approximation of the integral of F(x).
- Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.

- The problem y'=x+1, y(0)=1 has a dot table with x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375.
- The exact solution y=1/2+(x+1)^2/2 has values y=1, 1.28125, 1.625, 2.03125, 2.5000.
- Determine how the dot table was constructed and identify which rule [Rect, Trap, Simp] was applied.

Lecture on Euler, Heun, RK4 algorithms. Computer implementation. Geometric and algebraic ideas in the derivations.

Numerical work maple L3.2, L3.3, L3.4, L4.2, L4.3, L4.4 will be submitted after Fall Break.

- All discussion of maple programs will be based in the TA session [Fusi and Richins].
- There will be one additional presentation of maple lab details in the main lecture.

- y'=-2xy, y(0)=2, solution y=2exp(-x^2)
- y'=(1/2)(y-1)^2, y(0)=2, solution y=(x-4)/(x-2).

- y'=3x^2-1, y(0)=2, solution y=x^3-x+2
- y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x).
- y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
- y'=2x+1, y(0)=3 with solution y=x^2+x+3. [lecture notes only]

- Maple Labs 3 and 4, due after Fall Break.

Maple lab 3 F2009. Numerical DE (81.9 K, pdf, 19 Jul 2009)

Maple lab 4 F2009. Numerical DE (78.2 K, pdf, 19 Jul 2009)

- References for numerical methods:

How to use maple at home (3.8 K, txt, 10 Oct 2008)

Maple lab 3, problem 1 solution. Same as ER-1. (184.6 K, jpg, 08 Feb 2008)

F2008 notes on numerical DE report for Ch2 Ex 10 (34.8 K, pdf, 27 Aug 2008)

F2008 notes on numerical DE report for Ch2 Ex 12 (51.5 K, pdf, 27 Aug 2008)

F2008 notes on numerical DE report for Ch2 Ex 4 (34.4 K, pdf, 27 Aug 2008)

F2008 notes on numerical DE report for Ch2 Ex 6 (47.3 K, pdf, 27 Aug 2008)

Sample Report for 2.4-3 (342.9 K, pdf, 06 Feb 2005)

The work for 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4.

The numerical work using Euler, Heun, RK4 appears in L3.2, L3.3, L3.4.

Confused about what to put in your L3.2 report? Do the same as what appears in the sample report for 2.4-3.

In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...". Some browsers require SHIFT and then mouse-click. Open the saved file in xmaple or maple. Extension .mws or .mpl allow interchange between different versions of maple.

More details on maple lab 2.

How to present the solutions to the exam review problems ER-1 and ER-2. These are identical to math-only [no computer] problems L3.1, L4.1.

Lecture: 3.1, 3.2, frame sequences, combo, swap, multiply, plane and space geometry,

three possibilities, method of elimination, parameters,

Differential equations example and unique solution of a 2x2 system. See problem notes section 3.1:

Prepare 3.1 problems for next collection.

- Links for maple lab 2:

For more on superposition y=y_p + y_h, see Theorem 2 in the link

Linear DE part I (193.7 K, pdf, 27 Aug 2007)

For more about home heating models, read the following links.