MATH 24
ORDINARY DIFFERENTIAL EQUATIONS
Summer Session II, 2003



Updated detailed lecture schedule

Week 1
  • Basic definitions. Qualitative and Quantitative approaches.
  • Initial Value Problems. First order linear equations.
  • Method of Integrating factor.
  • Method of Variation of Parameters.
  • Integrating factor and first order linear equations (review).
  • Separable equations. Lots of exercises.
  • Modeling with first order equations. Mixing problem.
  • Quiz 1.
  • Remarks on direction fields.
  • Existence and uniqueness of solutions to initial value problems. Proof for the linear case.
  • Differences between linear and nonlinear first order differential equations.
Week 2
  • Autonomous equations. Interval of definition of a solution. Direction fields.
  • Population models. Stable and unstable equilibrium solutions.
  • Exact first order differential equations.
  • Integrating factors.
  • Numerical approximations. Euler's method.
  • First order difference equations. Stable, asymptotic stable, semistable and unstable equilibrium.
  • Stairstep diagrams (spider's web diagrams)
  • Discussion session on homework problems.
  • Second order equations. Linear equations.
  • Homogeneous equations with constant coefficients.
  • Characteristic equation. Positive, negative and zero discriminant cases. Several examples.
  • Quiz 2.
Week 3
  • Quick review of Linear Algebra. Vector space, linear combination, linear independence.
  • Span of a set, basis of a vector space, dimension, linear transformations.
  • The above concepts in the setting of second order linear homogeneous differential equations.
  • Review for the midterm.
  • The Wronskian.
  • Fundamental set of solutions.
  • Nonhomogeneous second order linear differential equations. Undetermined coefficients.
  • Variation of parameters (just mentioned). To be fully covered later when studying systems of
    first order linear differential equations.
  • Midterm.
Week 4
  • General theory of higher order linear equations.
  • Homogeneous equations with constant coefficients.
  • The method of undetermined coefficients.
  • Quiz 3.
  • Review of Matrices. Matrix functions.
  • Eigenvalues and eigenvectors.
  • Linear systems (dimension 2).
  • The matrix etAc as the general solution to the homogeneous system x'=Ax.
  • Linear systems. Homogeneous systems x'=A(t)x . Wronskian. Fundamental Matrix.
  • Nonhomogeneous linear systems x'=A(t)x+g(t). Variation of Parameters method.
  • Autonomous homogeneous linear systems x'=Ax. The matrix etA for different matrices A.
  • Quiz 4.
Week 5
  • The autonomous system x'=Ax with A in Mn×n with constant real entries.
  • The system X'=AX vs. x'=Ax. A fundamental matrix of X'=AX constructed with n
    linearly independent eigenvectors of A.
  • Homogeneous linear systems of dimension two. Phase portraits.
  • Classification of solutions to x'=Ax according to the sign of the discriminant tr2(A)-det(A).
    Case 1: det(A) not equal to zero. Case 2: det(A)=0.
  • Summary of classification of solutions of two dimensional linear homogeneous systems x'=Ax.
  • Bifurcation points.
  • Nonlinear homogeneous linear systems x'=v(x) (v(x) is a vector field different from Ax).
    Example: Predatory-prey model. Phase portrait analysis.
  • Review of nonhomogeneous linear systems x'=Ax+g(t). Variation of parameters.
  • Review
  • Final due: Fr, Aug 29
  • Students evaluations.