Systems of differential equations
Many problems involve several quantities changing together: predators and prey, coupled springs, chemical species. Systems of ODEs describe them all.
What a system is
A first-order system has several unknown functions, each with its own equation: \(x' = f(t, x, y)\), \(y' = g(t, x, y)\). The solution is a pair of functions, which can be plotted as a curve in the \(xy\)-plane, the phase plane. Initial conditions specify a starting point.
Converting higher-order equations
Any \(n\)th-order equation becomes a first-order system. For \(y'' + 3y' + 2y = 0\), let \(x_1 = y\) and \(x_2 = y'\). Then \(x_1' = x_2\) and \(x_2' = -2x_1 - 3x_2\). Numerical solvers work with first-order systems, so this conversion is used constantly.
Matrix form
A linear system with constant coefficients is written \(\mathbf{x}' = A\mathbf{x}\), where \(\mathbf{x}\) is a vector of unknowns and \(A\) a matrix. The example above has \(A = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix}\). Linear algebra now becomes the main tool.
\mathbf{x}' = A\mathbf{x}Eigenvalue solutions
Try \(\mathbf{x} = \mathbf{v}e^{\lambda t}\). Substituting gives \(A\mathbf{v} = \lambda\mathbf{v}\): \(\lambda\) must be an eigenvalue of \(A\) with eigenvector \(\mathbf{v}\). For the matrix above, \(\det(A - \lambda I) = \lambda^2 + 3\lambda + 2 = 0\) gives \(\lambda = -1, -2\), matching the characteristic roots of the original second-order equation.
A\mathbf{v} = \lambda\mathbf{v}The general solution
With distinct real eigenvalues \(\lambda_1, \lambda_2\) and eigenvectors \(\mathbf{v}_1, \mathbf{v}_2\), \(\mathbf{x} = c_1\mathbf{v}_1e^{\lambda_1t} + c_2\mathbf{v}_2e^{\lambda_2t}\). Each eigenvector is a straight-line solution direction in the phase plane, along which solutions simply grow or shrink exponentially.
Worked example
For \(A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}\), eigenvalues are 3 (eigenvector \((1, 1)\)) and \(-1\) (eigenvector \((1, -1)\)). So \(\mathbf{x} = c_1\begin{pmatrix}1\\1\end{pmatrix}e^{3t} + c_2\begin{pmatrix}1\\-1\end{pmatrix}e^{-t}\). Solutions along \((1, -1)\) shrink to the origin; all others eventually grow along \((1, 1)\).
Complex eigenvalues
If \(\lambda = \alpha \pm \beta i\), solutions spiral: they rotate with frequency \(\beta\) and grow or shrink like \(e^{\alpha t}\). Real solutions come from the real and imaginary parts of \(\mathbf{v}e^{\lambda t}\). Pure imaginary eigenvalues (\(\alpha = 0\)) give closed orbits, like an undamped spring.
Applications
Coupled tanks exchanging fluid, two masses joined by springs, competing species, and multi-loop circuits all lead to linear systems. In coupled oscillators, eigenvectors are the normal modes: patterns in which every part vibrates at the same frequency.
Key takeaways
Systems describe several interacting quantities; higher-order equations convert to first-order systems. Linear constant-coefficient systems \(\mathbf{x}' = A\mathbf{x}\) are solved with eigenvalues and eigenvectors. Real eigenvalues give exponential growth or decay along eigenvector lines; complex ones give spirals or closed orbits.
Flashcard
Several unknown functions, several equations
Check your understanding
The matrix A = [[2, 0], [0, -3]] has eigenvalues: