Week #1
Sept 24-28
  • Differential equations: what they are and why they are important; examples (heating and cooling, falling objects, population models); direction fields; equilibrium solutions; qualitative behavior (Section 1.1)
  • Linear differential equations of first order with constant coefficients (Section 1.1)
  • Linear differential equations of first order with variable coefficients; integrating factors; examples (Section 1.2)
    		•  Sep 24th: classes begin
    Week #2
    Oct 1-5
  • The method of separation of variables; examples of separable differential equations (Section 2.1)
  • More examples of separable equations (Section 2.1); the logistic growth model; autonomous differential equations (Section 2.4)
  • Homogeneous differential equations (cf. Problem 30 on page 51); examples; overview of the solution methods discussed so far
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    Week #3
    Oct 8-12
  • Exact differential equations (Section 2.5); examples
  • Exact differential equations (continued); separable equations as special cases of exact differential equations
  • Existence and uniqueness theory for ordinary differential equations (Section 2.3); the interval of existence of a solution; analysis of equilibria
    		•
    Oct 12th: Last day to add or drop a class.
    Week #4
    Oct 15-19
  • Systems of linear differential equations; coupled vs. uncoupled systems; applications to mixing problems; rewriting a differential equation of second order as a coupled system of two differential equations of first order; matrix notation (Section 3.2)
  • Existence and uniqueness theory for systems of linear ODE (Section 3.2); solving homogeneous systems with constant coefficients (Section 3.3); eigenvalues and eigenvectors
  • Solving homogeneous systems with constant coefficients (continued); examples; phase space plots; asymptotic behavior of solutions
    		•  Oct 18th: midterm I.
    Week #5
    Oct 22-26
  • Fundamental systems and Wronskian determinant (Section 3.3); 2x2 matrices with complex eigenvalues (Section 3.4); how to solve systems of ODE when the coefficient matrix has complex eigenvalues
  • How to solve systems of ODE when the coefficient matrix has complex eigenvalues (continued); expressing the solution in real form
  • More examples involving matrices with complex eigenvalues; matrices with repeated eigenvalues (Section 3.5)
    		•
    Week #6
    Oct 29 - Nov2
  • More examples involving matrices with repeated eigenvalues; asymptotic behavior of solutions; sinks, sources, and saddles; classification in terms of trace and determinant of the coefficient matrix (diagram on page 190)
  • An example of a 3x3 system of linear differential equations; inhomogeneous 2x2 systems; variation of parameters formula (Section 4.7)
  • Variation of parameters formula (continued); any two solutions to a given inhomogeneous equation differ by a solution to the homogeneous equation

    		•
    Week #7
    Nov 5-9
  • Second order linear differential equations; damped linear oscillations in mechanics (Section 4.4)
  • Linear oscillations in the presence of an external force; method of undetermined coefficients (Section 4.6)
  • Linear oscillations in the presence of an external force (continued); frequency response; gain function and phase shift; resonance (Section 4.6)
    		•
    Week #8
    Nov 12-16
  • Laplace transform: definition and basic properties; a formula for the Laplace transform of the n-th derivative of a function (Sections 5.1 and 5.2)
  • A formula for the Laplace transform of t * f(t); examples; solving differential equations using the Laplace transform (Section 5.4); the inverse Laplace transform (Section 5.3)
  • Solving differential equations using the Laplace transform (continued); computing the inverse Laplace transform of a rational function using partial fractions
    		•
    Nov 15th: Midterm II.
    Nov 16th: Change of grading basis deadline
    Course withdrawal deadline
    Nov 19-23 Thanksgiving recess
    Week #9
    Nov 26-30
  • Nonlinear systems of differential equations; existence and uniqueness theory (Section 3.6); autonomous systems and equilibria (Section 3.6, Example 1; Section 7.1)
  • Hamiltonian systems; the mathematical pendulum; conservation of energy (Section 7.1, Example 1)
    		•
    Week #10
    Dec 3-7
  • The mathematical pendulum (continued); predator-prey models (Section 7.4)
    Review and applications.
    		• End-Quarter Period.
    Final Exam December 10th 7-10PM