Differential Equations Midterm #3 Review

## 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 ertξ is a solution
• General solution: x = c1er1tξ(1) + c2er2tξ(2) + c3er3tξ(3) + ... + cnerntξ(n)
• Specific solutions can be found by substituting x(0) for x and 0 for t
• ...then solve the resulting system for c1, c2, ..., cn

• §7.6: Complex Eigenvalues
• Solution is still x = c1er1tξ(1) + c2er2tξ(2) + c3er3tξ(3) + ... + cnerntξ(n)
• If r is complex, use Euler's Formula to expand: e(a+bi)t = eat·(cos bt + i·sin bt)
• Combine entire general solution into one vector (eat 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 ertξ
• The other solution is tertξ + ertη, where η is another vector s.t. = 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 = c1ertξ + c2(tertξ + ertη)

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