Solving a Differential System

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Solving a Differential System Find the eigenpairs, build the general solution, then apply to pin down the constants. Solve the system of differential equations subject to the initial condition . Write it as , solve with eigenvalues and eigenvectors, and verify the answer satisfies both the system and the initial condition. The initial vector in eigen-coordinates The two dashed lines are the eigen-directions. The initial vector x (0) = (3, 1) (green) is the diagonal of the parallelogram built from its components 2v₁ and −1·v₂ — exactly the constants c₁ = 2 and c₂ = −1 we solve for below. Nicely done. You solved a linear system end to end with the eigenvalue method. Matrix form: writing the system as turns a coupled ODE into a linear-algebra problem. Eigenpairs are the building blocks: each pair gives a pure exponential solution . General solution: a linear combination . Constants from : setting t = 0 gives — here . Always verify: we confirmed and both hold. Because , both modes grow and the faster term dominates as — the trajectory bends toward the eigen-direction.

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