Systems of ODEs
Objective. Solve linear systems using eigenvalues.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Systems of ODEs describe several coupled quantities changing together, written as x′ = Ax for a vector x and constant matrix A. The method is to find eigenvalues λ and eigenvectors v of A, since each pair gives a solution e^(λt)v, and the general solution is a linear combination of these. For a 2×2 case with distinct real eigenvalues λ1, λ2, x(t) = c1 e^(λ1t) v1 + c2 e^(λ2t) v2, with c1, c2 fixed by initial conditions. Complex conjugate eigenvalues give oscillating solutions via Euler's formula, while repeated eigenvalues need a generalised eigenvector term t e^(λt)v.
Worked example
Solve the system of ODEs x′ = 3x, y′ = −2y with x(0) = 4, y(0) = 1.
- 1The system is already decoupled, each equation independent
- 2Solve x′ = 3x to get x(t) = x(0)e^(3t) = 4e^(3t)
- 3Solve y′ = −2y to get y(t) = y(0)e^(−2t) = e^(−2t)
- 4Combine the two solutions as the vector answer
Answer: x(t) = 4e^(3t), y(t) = e^(−2t)
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Practice
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Start practisingCommon mistake. A common error is assuming the general solution is just e^(At) applied to x(0) without ever finding eigenvalues, or mixing eigenvectors from different eigenvalues into one exponential term. Always pair each eigenvalue strictly with its own eigenvector.
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