Systems of Differential Equations and Eigenvalue Solutions
Several coupled unknowns become one vector equation. The eigenvalues of the coefficient matrix give the growth rates and the eigenvectors give the directions along which the system decouples.
What a learner can do afterwards
- Write a coupled pair as a single matrix equation
- Solve a two-by-two system through eigenvalues and eigenvectors
- Convert a second-order equation into a first-order system
1 · Read
When two quantities each feed the other's change, as sharks and sardines do, one equation cannot hold them. They become one vector equation, x prime equals A times x, with all four rates in the matrix A.
A diagonal matrix with two and three on its diagonal stretches one axis by two and the other by three. Its eigenvalues sit on the diagonal: two and three. Each axis direction is an eigenvector, since the matrix only stretches it. The vector one one is no eigenvector here: multiplying returns two three, which points another way.
For a general two by two system, subtract lambda on the diagonal and set the determinant to zero. That characteristic equation gives the growth rates, and each rate's hidden line gives its direction. To convert, name velocity v as y prime. For y double prime plus four y prime plus three y, the second line reads v prime equals minus three y minus four v.
Take rows 2 1 and 1 2. (2 minus lambda) squared minus 1 = 0 gives lambda 3 and 1. For 3, A minus 3I has rows -1 1 and 1 -1, so b = a: direction (1, 1). For 1, A minus I has rows 1 1 and 1 1, so b = -a: direction (1, -1). So x = c1 e^(3t) (1, 1) + c2 e^t (1, -1).
Pack the pair into a matrix, read growth from eigenvalues, direction from eigenvectors, and convert orders with a velocity variable.
Readings 'Solving a coupled pair', 'Why eigenvectors decouple a system' and 'A second-order equation as a system' are original to this dataset. All 5 extra questions are original to this dataset.
2 · Watch
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