Math 262 – Differential Equations, Fall 2011


Instructor: Anatolii Grinshpan
Office hours:  MT 6-7 or by appointment, Korman 253.

Syllabus   Terrell’s notes   Academic calendar    Math resource center   MIT video lectures

Sep 19. Introduction to the course.  First order equations. Slope fields.

              Slope field generator.


Sep 20. Stationary solutions. Solution curves and isoclines. Phase line.

             The exponential model. A gallery of equations.

             Homework 1: due September 27.


Sep 26. General solution, an initial-value problem. A glimpse into PDE: the transport equation.

             Population dynamics. Separation of variables.


Sep 27. Examples of separable equations. The leaking bucket.

             Existence and uniqueness of solutions. The maximal interval.

             Homework 2: due October 4.


Oct 3.   Slope fields of autonomous equations. Stability of stationary solutions.

             Change of variables: examples. Linear equations with constant coefficients.

             Integrating factors: examples. Euler’s method.


Oct 4.  Homework discussion.  A mixing model.

            Homework 3: due October 11.


Oct 10. No class (Columbus Day).


Oct 11. Homework discussion. Examples on Euler’s method.


Oct 14. Questions session: 6-7PM, Korman 245.


Oct 17. Midterm 1 (lectures of September 19 - October 11)


Oct 18. Linear equations: homogeneous and not. Second-order linear equations.

            Phase plane. Conservation of energy. Case of constant coefficients. Characteristic roots.

            Homework 4: due October 25.


Oct 24. Riccati’s equation and associated second-order linear equation.

             Distinct real characteristic roots. Repeated characteristic root.

             Purely imaginary characteristic roots. Simple harmonic oscillator.


Oct 25. Complex characteristic roots. Wronskian determinant and linear independence.

             Forced harmonic oscillator. The method of undetermined coefficients.  Resonance.

             Homework 5: due November 1.


Oct 31.  Harmonic oscillator revisited. The principle of superposition.

             The interval of linear dependence/independence. Abel's formula. Fundamental pairs.


Nov 1.  Averaging of sinusoids. Types of initial/boundary conditions.

             Examples on Abel’s formula. First look at linear systems.

             Homework 6: due November 8.


Nov 7.  Linear systems: mixing in interconnected tanks.

            Transition between first-order systems and second-order equations.


Nov 8.  Linear systems: multiple spring-mass problems.

            Nonlinear systems: predator-prey problems.


Nov 14. Discussion.


Nov 15. Midterm 2 (lectures of October 18–November 8)


Nov 21. Systems of linear equations with constant coefficients. Eigenvalues and eigenvectors.

              Distinct real eigenvalues. Nonreal eigenvalues. MATLAB examples.

              Phase portrait generator. John Polking’s pplane: MATLAB, JAVA.


Nov 22. Classification of phase portraits. Center. MATLAB examples.

              Homework 7: due November 29 (answers).


Nov 28. Phase portraits via trace and determinant.

              Phase portraits of nonlinear systems: predator-prey, van der Pol  (MATLAB examples).


Nov 29. Systems with eigendirection deficiency.

              Phase portraits of nonlinear systems: nonlinear oscillator, Lorenz (MATLAB examples).


Dec 5.  Questions session.


Dec 6.  Final Exam (all lectures).