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
Sharks and sardines in one sea, two water tanks joined by a pipe, your arms and the swing they push: when two quantities each feed the other's change, one equation cannot hold them. Two linked unknowns become one vector equation, x prime equals A times x. The matrix A holds all four rates in one place, and from here the whole system moves as a single object.
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.
Test any claimed eigenvector by multiplying: the result must run parallel to the claim. One one fails for the diagonal two three matrix, since it returns two three, which points elsewhere.
Pack the pair into a matrix, read growth from eigenvalues, direction from eigenvectors, and convert orders with a velocity variable.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.