# Differential Equations with Theory (18.034)

Ravi Vakil, Dept. of Mathematics Rm. 2-271, vakil@math.mit.edu.

FINAL EXAM AND COURSE GRADES ARE AVAILABLE HERE.

Remember to fill out a card with the various bits of information I asked for, and sign up to meet me. (Just ask if you have no idea what this is about.) If you missed the first recitation and class, be sure to e-mail me.

Lectures: Monday, Wednesday, Friday 1-2 in 2-146.

Recitations: Monday, Wednesday 10-11 in 2-146. If there is anything you would like to hear about in recitation, please e-mail suggestions (ideally two days in advance, but even the day before would be fine).

Office hours: Wednesday 11-12 and at one other hour in the week. You can also catch me after most recitations and classes.

Text: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, Sixth Edition. It should be available at Quantum Books. If you'd like something else to look at, you can check out M. Braun's Differential Equations and Their Applications. For a more beautiful and advanced (Russian) viewpoint, see Arnol'd's Ordinary Differential Equations.

Grading: Problem sets will be worth a total of 100, the two midterms (Mar. 10 and Apr. 14) will be worth 100 each, and the final exam will be worth 200.

Problem sets: The best 7 will count (out of an estimated 10, due on Fridays in class); we agreed that no lates will be accepted for any reason. Feel free to think about the problems in groups (collaboration is encouraged), but you must write up solutions on your own, and give credit for an idea to the person who thought of it. (You won't lose marks for doing so.)

Here is Problem Set 1 (in dvi, postscript, and pdf formats; the last one will most likely work for you).

Here is Problem Set 2 (dvi, ps, pdf), minus the figures for the last problem.

Here is Problem Set 3 (dvi, ps, pdf), minus the figure for the last problem.

Here is Problem Set 4 (dvi, ps, pdf).

Here is Problem Set 5 (dvi, ps, pdf), due Mar. 17. In problem 2, change the x's to t's.

Here is Problem Set 6 (dvi, ps, pdf), due Mar. 31.

Here is Problem Set 7 (dvi, ps, pdf), due Apr. 7.

Here is Problem Set 8 (dvi, ps, pdf), due Apr. 21.

Here is Problem Set 9 (dvi, ps, pdf), due Apr. 28.

Here is Problem Set 10 (dvi, ps, pdf), due May 5.

Here is Problem Set 11 (dvi, ps, pdf), consisting of Laplace transform practice problems, not to be handed in.

Hard copies (with figures) are available on the bulletin board beside my office door.

Midterms: There will be two midterms.

Here are some practice problems for Midterm 1, including a syllabus of the topics we've covered (dvi, ps, pdf). Here are answers to practice midterm 1.

Here is Midterm 1 (dvi, ps, pdf). Here are sketches of solutions (dvi, ps, pdf).

Here are some practice problems for Midterm 2, including a syllabus of the topics we've covered (dvi, ps, pdf). (A question on eigenvectors was added Apr. 12.) Here are answers to practice midterm 2.

Here is Midterm 2 (dvi, ps, pdf). Here are sketches of solutions (dvi, ps, pdf).

Final exam: Here are practice problems for the final (dvi, ps, pdf).

Here are (sketches of) solutions (minus figures) to the practice problems for the final (dvi, ps, pdf).

### Syllabus

A syllabus (covering what we've done) will appear here, most likely with a time lag of a few days. References are to the text.

Wed. Feb. 2 (Recitation). Recurrences, a discrete analogue of differential equations.

Wed. Feb. 2. Definitions: ODE, PDE, order, solution, linear ODE. Direction field. Solving y' = 6t+4, y'/y=1. Chapter 1.

Fri. Feb. 4. Integrating factors for first-order linear ODEs. Existence and uniqueness of solutions in that case. Separable equations. Sections 2.1-2.3.

Mon. Feb. 7 (Rec). Problems with differentials. The Brachistochrone problem. Autonomous equation preview.

Mon. Feb. 7. Integral curves. The existence and uniqueness theorem. Examples showing pathologies: y' = -t/y, y' = y^2, y' = y, y' = y^(1/3). Section 2.4. Autonomous equations (done quickly; make sure you understand the phase line, stable equilibria, unstable equilibria). Section 2.6.

Wed. Feb. 7 (Rec). Three problems: the logistic equation (Verhulst equation, see p. 70 #18), linear equations (p. 31 #26), Bernoulli equations (p. 33).

Wed. Feb. 9. Exact equations, Section 2.8. The "closed" necessary condition; sufficiency on a rectangle (or simply connected set). Where sufficiency fails on a non-simply connected set.

Fri. Feb. 11. Exact equations continued. The example (-y+xy')/(x^2+y^2) = d/dx arctan(y/x). Problem Set 1 due.

Mon. Feb. 14 (Rec). Homogeneous equations (Section 2.9): y' = f(x,y) where f(x,y) is a function purely of y/x, i.e. can be written F(y/x); then subtitute v=y/x. Orthogonal trajectories, e.g. to families x^2+y^2 = c^2, (x-c)^2+y^2=c^2, x^2-xy+y^2=c^2 (problems 2.10.36-38).

Mon. Feb. 14. Solutions to homogeneous linear ODEs with constant coefficients. The toy model: recurrences. The characteristic polynomial.

Wed. Feb. 16 (Rec). Linear algebra. Complex numbers.

Wed. Feb. 16. The case where the characteristic polynomial has distinct real roots. Repeated roots (without proof). Complex roots (with proof). Sections 3.1, 3.2, 4.2.

Fri. Feb. 18. Introduction to Linear operators (and linear equations with constant coefficients). Problem Set 2 due.

Tues. Feb. 22 (Rec). Solving y'''-2y''-y'+2y=0 (with initial conditions) "inductively". (Working through the proof to be given in class.)

Tues. Feb. 22. Proof on existence and uniqueness of solutions to homogeneous linear equations with constant coefficients.

Wed. Feb. 23 (Rec). PS2#7: When is a differential equation exact on a region R? Asides on delta functions, distributions, Lebesgue integration, etc.

Wed. Feb. 23. Solving nonhomogeneous linear differential equations with constant coefficients p(D) y = f(x): (i) find solutions to p(D)y = 0, (ii) find one solution to p(D)y = f(x) (by guessing = the method of undetermined coefficients, Section 3.6), and (iii) add the answers to (i) and (ii). A useful trick to guess well: If g(x) is a degree d polynomial, then (D-a) g(x) e^(bx) = h(x) e^(bx), where deg h = deg g -1 if a =b, and deg h = deg g otherwise. We did this when f(x) was an exponential, an exponential times a polynomial, and an exponential times a polynomial times a trig term. This is discussed in BD4.3; see also 4.1 and 4.2.

Fri. Feb. 25. Problem Set 3 due. Applying our general machine to second-order linear equations (Section 3.8).

Mon. Feb. 28 (Rec). Quantitative and qualitative behavior of y"+4y'+ay = 0 (playing with spring constant) when i) a=0, y(0)=-3, y'(0)=16. ii) a=3, y(0)=1, y'(0)=-5. iii) a=4, y(0)=2, y'(0)=-3. iv) a=5, y(0)=0, y'(0)=1.

Mon. Feb. 28. Further analysis of m y" + gamma y' + k y = 0 when gamma is very small (BD3.9) . Damping decreases frequency. Forced vibrations: m y" + ky = F_0 cos wt (no damping). If w <> root(k/m)=w_0, then general solution is R cos(w_0 t - delta) + (F_0/ ( m (w_0^2-w^2))) cos wt. If y(0)=y'(0)=0, get beats if w and w_0 are very close: y = (F/ (m (w_0^2 - w^2))) (cos wt - cos w_0 t). cos a - cos b = 2 sin ((a+b)/2) sin ((a-b)/2).

Wed. Mar. 1 (Rec). Beats (BD3.9). Simplify sin at - sin bt. Qualitative behavior of cos t + epsilon sin t. Strobe lights. Frequency and music.

Wed. Mar. 1. Resonance: m y" + ky = F_0 cos wt, w=root(k/m). Adding damping: m y" + gamma y' + k y = F_0 cos wt. The amplitude drops, and there is a phase lag (either a little more than 0 if w is smaller than w_0, or a little less than 180 if w is greater than w_0). BD3.9.

Fri. Mar. 3. Fored vibrations with damping, BD 3.9. Problem Set 4 due. Practice midterm 1 (and syllabus for midterm) out; see above to get it from this page.

Mon. Mar. 6 (Rec). Several questions, including an introduction to the Wronskian.

Mon. Mar. 6. Second-order equations without constant coefficients. Introduction to the Wronskian. BD3.2 and 3.3.

Wed. Mar. 8 (Rec). Abel's theorem. Wronskian practice.

Wed. Mar. 8. BD Section 3.3, including Abel's theorem. Application: using the Wronskian to find one solution to an order 2 homogeneous equation given another.

Fri. Mar. 10. First Midterm (in class).

Mon. Mar. 13 (Rec). Finding W(af+bg, cf+dg) given W(f,g), where a, b, c, d are constants. Using the Wronskian to find one solution to an order 2 homogeneous equation given another. Using "variation of parameters" to solve the following problem: (BD Ex. 4.1.26) given one solution y_1 to y''' + p_1 y'' + p_2 y' + p_3 y = 0, find a second-order equation from which you can find the rest of the solutions.

Mon. Mar. 13. Variation of parameters, BD3.7.

Wed. Mar. 15 (Rec). Variation of paramaters: generalizing downwards (to first-order equations).

Wed. Mar. 15. Variation of parameters for higher-order equations, BD4.4.

Fri. Mar. 17. Problem Set 5 due. Differential equations and physics.

Mon. Mar. 27 (Rec). Existence and uniqueness theorems, and topology.

Mon. Mar. 27. Philosophy of proof of the Existence and Uniqueness Theorem for Frist-order Equations (BD2.11). The Contraction Mapping Theorem.

Wed. Mar. 29 (Rec). More on topology and existence theorems: fixed-point theorems and applications.

Wed. Mar. 29. Using the Contraction Mapping Theorem to compute things such as root(2). Beginning the proof of the Existence and Uniqueness Theorem: translating the equation into integral form; Picard iterates.

Fri. Mar. 31. Continuing the proof of the Existence and Uniqueness Theorem: the Lipschitz condition, and Lipschitz constants. Problem Set 6 due.

Mon. Apr. 3 (Rec). The existence and uniqueness theorem applied to z'=z: the exponential function e^x exists.

Mon. Apr. 3. Finishing the proof of the existence and uniqueness theorem.

Wed. Apr. 5 (Rec). Systems of first-order linear differential equations, e.g. x'=y, y'=-x. Phase portrait, direction fields. Practice sketching.

Wed. Apr. 5. Introduction to systems of first-order differential equations (7.1). Existence and uniqueness statements (Thm. 7.1.1, 7.1.2). Lightning introduction to linear algebra (7.2, 7.3).

Fri. Apr. 7. Problem Set 7 due. Qualitative analysis of systems of equations. Guest lecturer (Prof. Marcolli).

Mon. Apr. 10 (Rec). More examples of systems of linear first-order equations.

Mon. Apr. 10. Basic theory of systems of first-order linear equations (7.4). Principle of superposition (7.4.1). Wronskian. Fundamental set of solutions. Beginning study of systems with constant coefficients.

Wed. Apr. 12 (Rec). Review of Wronskians. Computing powers of matrices with distinct eigenvalues.

Wed. Apr. 12. Review for midterm. How to solve a system of differential equations with (i) distinct eigenvalues, (ii) repeated eigenvalues. A handout on Phase portraits in two dimensions (by Prof. Miller) is available outside my office.

Fri. Apr. 14. Second Midterm (in class).

Wed. Apr. 19 (Rec). Conservation laws.

Wed. Apr. 19. Solving a system of linear differential equations with complex eigenvalues.

Fri. Apr. 21. Problem set 8 due. Fundamental mtarices, BD 7.8. Power series introduction, BD 5.1.

Mon. Apr. 24 (Rec). Conservation laws and i^i. Fun with power series.

Mon. Apr. 24. Power series continued.

Wed. Apr. 26 (Rec). Fun with power series continued.

Wed. Apr. 26. Series solutions near an ordinary point, BD 5.2-5.3.

Fri. Apr. 28. Initial ideas on what to do near a regular singular point, BD 5.4. Problem set 9 due.

Mon. May 1 (Rec). Rationality and irrationality; algebraicity and transcendence. Why root(2), e, and pi are irrational. Some comments on why e is transcendental.

Mon. May 1. Series solutions near a regular singular point, BD 5.4-5.7.

Wed. May 3 (Rec). The gamma function. (1/2)! = root(pi)/2. What power series tell you about combinatorics.

Wed. May 3. The Laplace transform.

Fri. May 5. The Laplace transform (up to 6.4). Problem set 10 due.

Mon. May 8. Guest lecture (Prof. Marcolli). The delta function and impulse functions (6.5). Remarks on the theory of distributions and generalized functions.

Wed. May 10 (Rec). More on the delta function. Coffee in Walker.

Wed. May 10. The convolution integral (6.6).

Tues. May 16. Final Exam, 1:30-4:30, Rm. 4-159.