12-267

DBN: Classes: 2012-13: 12-267 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class.  Monday: Introduction and the Brachistochrone.  Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes.  Friday: Some basic techniques: first order linear equations. 2 Sep 17  Monday: Separated equations, escape velocities. HW1.  Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations.  Friday: Reverse engineering separated and exact equations. 3 Sep 24  Monday: Solving exact equations, integration factors. HW2.  Tuesday: Statement of the Fundamental Theorem. Class Photo.  Friday: Proof of the Fundamental Theorem. 4 Oct 1  Monday: Last notes on the fundamental theorem. HW3.  Tuesday Hour 1: The chain law, examples of variational problems.  Tuesday Hour 2: Deriving Euler-Lagrange.  Friday: Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving.  Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4.  Friday: More Lagrange multipliers, numerical methods. 6 Oct 15  Monday: Euler and improved Euler.  Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods.  Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22  Monday: Multiple roots, reduction of order, undetermined coefficients.  Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29  Monday: The basic properties of matrix exponentiation.  Tuesday: Matrix exponentiation: examples.  Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5  Monday: Non-homogeneous systems.  Tuesday: The Catalan numbers, power series, and ODEs.  Friday: Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7.  Friday: Series solutions for $y'=f(x,y)$. 11 Nov 19  Monday: $\pi$ is irrational, more on the radius of convergence.  Tuesday (class): Airy's equation, Fuchs' theorem.  Tuesday (tutorial): Regular singular points. HW8.  Friday: Discussion of regular singular points.. 12 Nov 26  Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF).  Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9.  Friday: Non-oscillation, Sturm comparison. 13 Dec 3  Monday: More Sturm comparisons, changing the independent variable.  Tuesday: Amplitudes of oscillations. Last class was on Tuesday! F1 Dec 10 F2 Dec 17 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

Advanced Ordinary Differential Equations

Department of Mathematics, University of Toronto, Fall 2012

Agenda: If calculus is about change, differential equations are the equations governing change. We'll learn much about these, and nothing's more important!

Instructor: Dror Bar-Natan, drorbn@math.toronto.edu, Bahen 6178, 416-946-5438. Office hours: by appointment.

Classes: Mondays, Tuesdays, and Fridays 9-10 in RW 229.

Teaching Assistant: Jordan Bell, jordan.bell@utoronto.ca. Tutorials: Tuesdays 10-11 at RW 229. No tutorials on the first week of classes.

Text

Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).

Further Resources

• Undergraduate Information at the UofT Math Department

• Undergraduate Course Descriptions.

• Vitali Kapovitch's 2007 classes: Spring, Fall.

• Also previously taught by T. Bloom, C. Pugh, D. Remenik.

• My 12-267 notebook.

Dror's notes above / Student's notes below

Solving the complicated integral in the Brachistochroe integral

integral sqrt((d-y)/y) dy
=  integral sqrt(d y-y^2)/y dy

For the integrand sqrt(d y-y^2)/y, complete the square:

=  integral sqrt(d^2/4-(y-d/2)^2)/y dy

For the integrand sqrt(d^2/4-(y-d/2)^2)/y, substitute u = y-d/2 and du = dy:

=  integral (2 sqrt(d^2/4-u^2))/(d+2 u) du
= 2 integral sqrt(d^2/4-u^2)/(d+2 u) du

For the integrand sqrt(d^2/4-u^2)/(d+2 u), (assuming all variables are positive) substitute u = 1/2 d sin(s) and du = 1/2 d cos(s) ds. Then sqrt(d^2/4-u^2) = sqrt(d^2/4-1/4 d^2 sin^2(s)) = 1/2 d cos(s) and s = sin^(-1)((2 u)/d):

= d^2/2 integral (cos^2(s))/(d sin(s)+d) ds

For the integrand (cos^2(s))/(d sin(s)+d), substitute p = tan(s/2) and dp = 1/2 sec^2(s/2) ds. Then transform the integrand using the substitutions sin(s) = (2 p)/(p^2+1), cos(s) = (1-p^2)/(p^2+1) and ds = (2 dp)/(p^2+1):

= d^2/2 integral (2 (1-p^2)^2)/((p^2+1)^3 ((2 d p)/(p^2+1)+d)) dp

Simplify the integrand (2 (1-p^2)^2)/((p^2+1)^3 ((2 d p)/(p^2+1)+d)) to get (2 (p-1)^2)/(d p^4+2 d p^2+d):

= d^2/2 integral (2 (p-1)^2)/(d p^4+2 d p^2+d) dp
= d^2  integral (p-1)^2/(d p^4+2 d p^2+d) dp
= d^2  integral (p-1)^2/(d (p^2+1)^2) dp
= d integral (p-1)^2/(p^2+1)^2 dp

For the integrand (p-1)^2/(p^2+1)^2, use partial fractions:

= d integral (1/(p^2+1)-(2 p)/(p^2+1)^2) dp
= d integral 1/(p^2+1) dp-2 d integral p/(p^2+1)^2 dp

For the integrand p/(p^2+1)^2, substitute w = p^2+1 and dw = 2 p dp:

= d integral 1/(p^2+1) dp-d integral 1/w^2 dw

The integral of 1/(p^2+1) is tan^(-1)(p):

= d tan^(-1)(p)-d integral 1/w^2 dw
= d tan^(-1)(p)+d/w+constant

Substitute back for w = p^2+1:

= (d ((p^2+1) tan^(-1)(p)+1))/(p^2+1)+C

Substitute back for p = tan(s/2):

= 1/2 d (cos(s)+2 tan^(-1)(tan(s/2))+1)+C

Substitute back for s = sin^(-1)((2 u)/d):

= 1/2 (sqrt(d^2-4 u^2)+2 d tan^(-1)((2 u)/(d (sqrt(1-(4 u^2)/d^2)+1)))+d)+C

Substitute back for u = y-d/2:

= d (-tan^(-1)((d-2 y)/(2 d sqrt((y (d-y))/d^2)+d)))+sqrt(y (d-y))+d/2+C

Factor the answer a different way:

= 1/2 (-2 d tan^(-1)((d-2 y)/(2 d sqrt((y (d-y))/d^2)+d))+2 sqrt(y (d-y))+d)+C

Which is equivalent for restricted y and d values to:

= y sqrt(d/y-1)-1/2 d tan^(-1)((sqrt(d/y-1) (d-2 y))/(2 (d-y)))+C Syjytg 23:00, 11 September 2012 (EDT)