Exam Review Topics

• Differential Operators
  • Proving that an operator is linear
  • Factoring operators and reducing to a system of first order equations
  • General solution to nth-order homogeneous constant coefficient equations (Theorem 4.4 on page 646 of the Crowell and Slesnick handout)
• Linear Independence
  • Generic definition
  • Wronskian
  • fundamental sets of solutions
• Non-homogeneous equations
  • Solutions look like y(t) = y_h(t) + y_p(t) where y_h(t) is a solution to the homogeneous equation, and y_p(t) is a particular solution of the D.E.
  • Undetermined Coefficients
  • Variation of Parameters
  • Reduction of Order
• Vector Spaces
  • linear independence
  • basis
  • dimension
  • solutions to homogeneous equations
• Mechanical Vibrations
  • Free vibrations
    • Undamped
    • Damped
  • Forced vibrations
    • Undamped
      • Beats
      • Resonance
    • Damped
      • transient solutions
      • steady-state solutions

Important Theorems:

• Theorem 3.2.2 Principle of Superposition
• Theorem 3.3.1 Linear Independence and the Wronskian (Non-D.E. case)
• Theorem 3.3.2 Abel's Theorem (computing the Wronskian)
• Theorem 3.3.3 Linear Independence and the Wronskian (for solutions of a homogeneous linear D.E.)
• Theorem 4.1.2 Linear independence of solutions for nth order linear D.E.'s

Stuff you should know about solving first and second order linear D.E.'s (a.k.a. things to know from the first exam's material)

• integrating factors
• characteristic equations
  • distinct real roots
  • repeated roots
  • complex roots
• separable equations