MATH 351, Spring 2010 Course topics

1. Differential equations and models
   1. Equations and solutions. (1.1.1)
   2. Geometrical interpretation. Direction fields (1.1.2)
   3. Pure time equations (1.2)
   4. Mathematical models: mass on a spring (1.3)
2. First order equations
   1. Separable equations. Homogeneous equations (2.1)
   2. First-order linear equations (2.2)
   3. Picard iteration (2.3.1)
   4. Euler's method (2.3.2)
3. Second order linear equations
   1. Examples: mass on a spring; electrical circuits (1.3.6, Example 1.14, pp.24-25)
   2. Homogeneous linear equations with constant coefficients (3.2)
   3. Initial-value problem, existence/uniqueness; superposition principle, general solution, linear independence and the Wronskian (lecture notes, or Sections 3.2, 3.3, also Boyce-DiPrima, Theorem 3.3.2) (3.2)
   4. Inhomogeneous linear equations with constant coefficients. Method of undetermined coefficients (3.3)
   5. Cauchy-Euler equations (3.4.1)
   6. Power series solutions for equations with variable coefficients (3.4.2)
   7. Reduction of order (3.4.3)
   8. Boundary-value problems (Examples 3.22, 3.23 in 3.5)
   9. Higher-order equations (3.6)
4. Systems of linear equations
   1. Examples: mass-spring systems, electrical circuits, population dynamics
   2. Solutions, orbits and the phase plane (5.1, 5.3.1)
   3. The eigenvalue problem (5.3.2)
   4. Types of phase portraits depending on eigenvector/eigenvalue structure (5.3.3-5)
   5. Equilibrium solutions and stability (5.3.6)
   6. Nonhomogeneous systems: variation of parameters and undetermined coefficients (5.4)
   7. Three-dimensional systems (5.5)
5. Nonlinear systems
   1. Phase plane; examples (6.1, 6.3)
   2. The Lotka-Volterra system (6.1.2)
   3. Linearization and stability (6.3)