MATH 215: Ordinary Differential Equations

Quiz 1

• integrable and separable equations
  • separable first order ODEs
• IVP
• Slope field
• PICARD Theorem
  • internal of existence
  • unique
• ODE Classification
• Integration Factor
• Autonomous equation
  • critical points
  • stable
• Exact equation

Quiz 2

• 2nd order linear equations
  • homogeneous
  • theorem of uniqueness and existence
  • method of reduction order
  • 2nd order ODE with constant coefficients
    • characteristic equation
      • two real roots form
      • repeated root form
      • two complex root form
  • mechanical vibrations
    • free: undamped, unforced
      • change to positive
      • arctan
    • damped free vibrations
      • practical resonance
      • overdamped, two real
      • critical damped, repeated
      • under damp, two complex
  • nonhomogenous 2nd order ODE with constant coefficients
    • y = y_c + y_p
      • y_c
      • y_p
    • method of undetermined coefficients
      • find y_c
      • guess y_p
        • form of y_p must not match any term in y_c
      • determine free coefficients
      • sum of y_p
    • general solution = y_c + sum y_p
    • find C₁ and C₂

Quiz 3

• laplace transformations
  • shifting rule 1
  • shifting rule 2
  • heaviside function
  • laplace transformation of integrals
  • convolution
  • solving ODEs with laplace transformations
    • laplace transformation of derivatives

Quiz 4

• systems of linear equations
  • eigenvalue method: finding eigenvalues of 2by2 and 3by3
    • real eigenvalues
    • complex eigenvalues
    • repeated eigenvalues
      • multiplicity:
        • algerbraic
        • geometric
          • solve (P − λI)v₂ = v₁
          • general solution: x(t) = C₁v₁e^λ₁t + C₂(tv₁ + v₂)e^λ₁t

final

• 2d systems
  • features:
    • critical points
    • eigenvalue/eigenvector => eigendirection
    • nullcline
    • trajection
• vector fields
• stability
• different cases for linear systems

non linear systems

$$\vec{x^{,}} = Px + f$$