Midterm III Topics

• Series Solution of Differential Equations - Section 8.4
  • Determining singular points and characterizing them as regular or irregular - Section 8.4
  • Using the method of Frobenius to determine a power series solution near a regular singular point - Section 8.4
• First Order Linear Systems of Differential Equations - Sections 4.1-4.7
  • Theory of first order systems - Sections 4.1-4.4
    • Existence and uniqueness of solutions
    • Fundamental sets of solutions
    • The fundamental matrix
    • The Wronskian and its use to verify that solutions form a fundamental set
    • The superposition principle
    • Abel's theorem and its consequences
  • Writing n^th order linear differential equations as a linear system - Section 4.2
  • Writing a linear system of differential equations in matrix form - Sections 4.2-4.3
  • Constant coefficient homogeneous systems - Sections 4.5-4.7
    • Determining eigenvalues and corresponding eigenvectors (i.e., an eigenpair) for a 2x2 matrix
    • Finding general solutions and solving IVP's of a 2x2 system using eigenpairs for
      1. distinct eigenvalues - Section 4.5
      2. complex eigenvalues - Section 4.6
      3. repeated eigenvalue (with algebraic multiplicity equal to geometric multiplicity or algebraic multiplicity larger than geometric multiplicity) - Section 4.7
• Nonlinear systems - Sections 6.1-6.5; class notes
  • Determining equilibrium solutions of an autonomous system - Sections 6.1-6.2
  • Drawing the phase portrait for a conservative autonomous system - Section 6.3 and class notes
  • Stability - Section 6.4
  • Stability characteristics of y' = Ay - Section 6.4
  • Linearization and stability of nonlinear systems - Section 6.5