Differential Equations Midterm #3 Review

• §7.2, §7.3: Linear Algebra Review
  • Solving a system of equations with augmented matrices
    • ...even if the system is underdetermined (many solutions)
  • Definition of linear independence
  • Checking for linear independence using the Wronskian determinant
  • Eigenvalues & eigenvectors: definition and how to find them


• §7.1, §7.4: Basic Theory of Systems of Differential Equations
  • Any nth order differential equation can be rewritten as a system of n 1st order equations
  • Meaning of linear vs. nonlinear & homogeneous vs. nonhomogeneous for systems
  • A system of linear differential equations can be written as a single matrix equation
  • A homogeneous system of n equations should have exactly n independent solutions
    • Any linear combination of these solutions is also a solution


• §7.5: Solving Homogeneous Linear Systems with Constant Coefficients
  • Can be written as x' = Ax
  • To solve, find eigenvalues and eigenvectors of A
    • If r is an eigenvalue with eigenvector ξ, then e^rtξ is a solution
    • General solution: x = c₁e^r₁tξ⁽¹⁾ + c₂e^r₂tξ⁽²⁾ + c₃e^r₃tξ⁽³⁾ + ... + c_ne^r_ntξ⁽ⁿ⁾
  • Specific solutions can be found by substituting x(0) for x and 0 for t
    • ...then solve the resulting system for c₁, c₂, ..., c_n


• §7.6: Complex Eigenvalues
  • Solution is still x = c₁e^r₁tξ⁽¹⁾ + c₂e^r₂tξ⁽²⁾ + c₃e^r₃tξ⁽³⁾ + ... + c_ne^r_ntξ⁽ⁿ⁾
  • If r is complex, use Euler's Formula to expand: e^(a+bi)t = e^at·(cos bt + i·sin bt)
  • Combine entire general solution into one vector (e^at can remain outside)
  • Split into a vector of real terms plus a vector of imaginary terms
  • Factor (c₁ + c₂) out of the real vector and (c₁ - c₂)i out of the imaginary vector
  • Replace (c₁ + c₂) and (c₁ - c₂)i with C₁ and C₂ for pure real solution


• §7.8: Repeated Eigenvalues
  • A repeated eigenvalue isn't a problem as long as it produces distinct eigenvectors
  • Repeated eigenvectors, however, call for a different approach
  • Suppose r is the (only) eigenvalue and ξ is the (only) eigenvector
  • Clearly one of the solutions is e^rtξ
  • The other solution is te^rtξ + e^rtη, where η is another vector s.t. Aη = rη + ξ
  • To find η, solve (A - rI)η = ξ with an augmented matrix (you know r and ξ already)
  • η can be written as a multiple of ξ plus another vector; the multiple of ξ can be ignored
  • General solution: x = c₁e^rtξ + c₂(te^rtξ + e^rtη)

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