Math 20D - Introduction to Differential Equations - Fall 2008

| General Info | Calendar | Announcements | Additional Help | Homework | Matlab Homework | Practice Exams | Lecture Summaries|

General Information

Meeting TimeMon., Wed., Fri., 10:00 - 10:50
Location LEDDN AUD
Instructor Dragos Oprea
Lab Sections are held in WLH 2206 Directions.
You will need to go to the lab led by the TA who teaches your section.
  • B50, Tu 8-8:50
  • B51, Tu 9-9:50
  • B52, Tu 10-10:50
  • B53, Tu 11-11:50
  • B54, Tu 12-12:50
  • B55, Tu 1-1:50.
Textbook Elementary Differential Equations, ninth edition, by William E. Boyce and Richard C. DiPrima; published by John Wiley & Sons, Inc.;
Required and available at the bookstore and on reserve in the library. We will cover parts of Chapters 1, 2, 3, 5, 6 and 7 of the text.
The grade is computed as the best of the following weighed averages:
  • Homework 10%, Matlab 10%, Midterm I 20%, Midterm II 20%, Final Exam 40%.
  • Homework 10%, Matlab 10%, Midterm I 15 %, Midterm II 15%, Final Exam 50%.
In addition, you must pass the final examination in order to pass the course.
Ordinary differential equations: exact, separable, and linear; constant coefficients, undetermined coefficients. Variations of parameters. Series solutions. Systems. Laplace transforms. Techniques for engineering sciences. Computing symbolic and graphical solutions using Matlab.
Prerequisites Math 20C (or Math 21C) with a grade of C- or better.
ReadingsReading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.
Calculators Calculator use will not be permitted on exams. Graphing calculators are not required for the course.
Homework Homework problems will be assigned on the course homework page. You may drop the homework in the drop box on the 6th floor of AP&M by 3:50PM on Friday.

You may work together with your classmates on your homework and/or ask the instructors, the TA's, or tutors in the calculus lab for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.

Please adhere to the following neatness guidelines for homework that you turn in to be graded.
  • Write your name and section clearly on the front page of your completed assignment.
  • Clearly number each solution.
  • Write clearly and legibly.
Matlab homework is due Thursdays during section or else in the drop-off box on the 6th floor of AP&M at 4:50PM on Thursday. The Matlab homework page is here.
There will be two midterm exams given in class. The dates are Oct 19 and Nov 18. There will be no makeup exams.
The final examination will be held on Friday, December 9, 8-11AM. There is no make up final examination.
It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.
Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university.

Announcements & Dates

Important Dates and Class Holidays:
  • Friday, September 23: First lecture.
  • Friday, October 7: Add deadline.
  • Wednesday, October 19, Midterm I.
  • Friday, October 21: Drop deadline.
  • Friday, Nov 11: Veterans' Day.
  • Friday, November 18, Midterm II.
  • Thursday-Friday November 24-25: Thanksgiving Recess -- No Class
  • Friday, December 9: FINAL EXAM, 8-11am.

Final Exam Information

The final exam is Friday, December 9, 8-11am. You can bring a "cheat sheet" which should be at most a page.

Topics for the final.

Practice exams can be found at here.

Solutions to John Hall's practice final: page 1 , 2 , 3 , 4 , 5ab , 5cd , 6 , 7 , 8a, 8b, 8c .

Dragos's office ours for final exam: Tue 4-5, Wed 3-5 in APM6101.

Chris's office hours: Tue 1-3, Wed 1-3, APM 6446.

Allison will be away this week, but if you are enrolled in her section, you can pick up your old homeworks from Chris.

Midterm II Information

The midterm is Friday, November 21, 12-12:50 in WLH 2005. You are allowed to have a "cheat sheet" which should be at most a page (front only).

Review for Midterm II will be on Wednesday during lectures. Bring questions!

Practice midterms can be found at here.

The midterm topics.

Midterm I Information

The midterm is Wednesday, October 22, 12-12:50. You are allowed to have a "cheat sheet" which should be at most half a page back and forth.

The midterm topics.

We will take the midterm in 3 different rooms as follows:

  • Room WLH2005 if the first letter of your last is A-La.
  • Room CENTR 105 if the first letter of your last name is Le-R.
  • Room CENTR 109 if the first letter of your last name is S-Z.

Interactive campus map .

Because of the midterm, the 4th Matlab is due on Friday at 3:50PM.

Review for Midterm I will be on Monday during lectures. Bring questions!

Practice midterms can be found at here.

Additional Help

If you are having trouble with the homework or have questions about the material, the best way to get help is to attend the office hours offered by me and the teaching assistants. If you can't make the scheduled times, then email us and we'll set up an appointment.

Additional help is given by

  • Calculus Tutoring
  • Calculus Online Community.
  • Calculus Review
  • Lecture Summaries

    Lecture 1: Introduction to differential equations. Savings account modelling. Classification of differential equations: ordinary/partial, linear/nonlinear, order of an ODE.

    Lecture 2: Direction fields. Integral curves. Equilibrium solution. I solved the equation dy/dx=1/10 y-1 geometrically, showed that solutions diverge from the equilibrium solution. I also solved the equation analytically by separating variables and integrating, and analyzed the behaviour of solutions.

    Lecture 3: Linear 1st order ODE. Standard linear form. Integrating factors. I solved a couple of examples via integrating factors.

    Lecture 4: Modelling mixture problems. Existence and uniqueness of solutions for linear first order ODEs. The case of continuous vs. discontinuous coefficients. Integral curves don't cross.

    Lecture 5: Nonlinear 1st order ODEs. Separation of variables. Solutions may not be unique and they may have discontinuities which are not visible from the original equation.

    Lecture 6: Autonomous ODEs. Critical points. Phase line. Types of critical points: asymptotically stable, unstable, semistable. Example: population growth, logistic growth.

    Lecture 7: Exact 1st order differential equations: M+Ny'=0. How to check an equation is exact (M_y=N_x). Finding the function whose derivatives are M and N (e.g. f_x=M, f_y=N). This function f is constant along solutions. Examples.

    Lecture 8: Linear second order ODE and IVP. Superposition of solutions. Existence and uniqueness of solutions. The case of constant coefficients. The characteristic equation. Examples.

    Lecture 9: Wronskian for two solutions (y_1, y_2) is given as a 2 x 2 determinant. Nonzero wronskian means fundamental set of solutions (which means that the general solution is of the form y=c_1y_1+c_2 y_2). Abel's theorem shows how to compute the Wronskian.

    Lecture 10: Constant coefficient 2nd order ODE. The case of complex roots. Complex exponentials. The fundamental pair is obtained by taking the real and the imaginary part of the exponential.

    Lecture 11: Midterm review. I announced the midterm rooms, midterm topics, solved a few sample problems.

    Lecture 12: The case of repeated roots r_1=r_2=a. A fundamental pair of solutions is y_1=e^{at}, y_2=te^{at}. Non-homogeneous equations have solutions y=y_p+y_h where y_p is a particular solution and y_h solves the homogeneous equation. I showed an example where the particular solution is found by method of undetermined coefficients (using polynomials).

    Lecture 13: More undetermined coefficients. I showed the examples involving trigonometric functions, exponentials, and combinations of such.

    Lecture 14: Variation of parameters. I showed how to look for solutions of the form y=u_1 y_1+u_2 y_2, where y_1, y_2 solve the homogeneous equation and u_1, u_2 are functions to be determined. I found the expression of u_1, u_2. Examples.

    Lecture 15: Linear first order systems of ODEs. I showed how a first order linear system becomes a second order differetial equation and conversely. I solved a system which came from modelling temperature. Matrices: multiplication, multiplication by vectors, inverses.

    Lecture 16: Using matrices to solve systems. Homogeneous systems admit nontrivial solutions if the matrix of coefficients has zero determinant. Eigenvalues and eigenvectors. Eigenvalues are found by solving det(A-lambda I)=0.

    Lecture 17: Eigenvalues are found by solving the characteristic equation. Linear independence and basis. Check for basis by computing the determinant of the matrix whose columns are the given vectors.

    Lecture 18: Solving first order systems of ODEs X'=AX. Find two solutions in the form e^{lambda t} v, where lambda is an eigenvalue of A, and v is an eigenvector. Superposition of solutions. Linear independence and Wronskian. Sketching solutions: saddle (eigenvalues of opposite signs), sink nodes (negative eigenvalues), source nodes (positive eigenvalues).

    Lecture 19: Complex eigenvalues. Take the real and imaginary parts of the complex solutions. The trajectories are spirals. To find the direction (clockwise vs. counterclockwise) pick a point on the spiral and find the velocity vector. I started repeated eigenvalues.

    Lecture 20: Repeated eigenvalues and improper nodes. To find the direction of the trajectories, compute the velocity of one vector at a point. Fundamental matrix Psi(t) and the normalized fundamental matrix Phi(t)=Psi(t) Psi(0)^-1. Solving IVP: X=Phi(t)X_0.

    Lecture 21: Exponential of a matrix. The normalized fundamental matrix is Phi(t)=e^{At}. Diagonalization.

    Lecture 22: Diagonalization continued T^{-1}AT=D, D=diagonal with eigenvalues on the diagonal. Two applications: decoupling systems (even when they are not homogeneous) x'=Ax+b. Look for x=Ty, then y'=Dy+T^{-1}b. Second application: compute exponentials e^{At}=Te^{Dt}T^{-1}.

    Lecture 22: Midterm review.

    Lecture 23: Radius of convergence of power series. Series solutions near ordinary points. I solved y''+y=0, found the recursion for the series coefficients.

    Lecture 24: Laplace transform. Examples: Laplace transform of 1, e^{at}, sin (at), cos(at), t^n. Inverse Laplace transform. Partial fractions.

    Lecture 25: Laplace transform of derivatives. Solving ODEs using Laplace transform.

    Lecture 26: Step functions. Laplace transform of step functions. The inverse Laplace transform of e^{-sa}F(s) is u_a(t)f(t-a). I solved a differential equation involving step functions.