Nonhomogeneous systems and applications
When outside inputs drive a system, its response combines the natural modes with a particular response, just as for single equations.
Structure
For \(\mathbf{x}' = A\mathbf{x} + \mathbf{g}(t)\), the general solution is \(\mathbf{x} = \mathbf{x}_h + \mathbf{x}_p\): homogeneous plus particular. The homogeneous part uses eigenvalues and eigenvectors; the particular part responds to the forcing.
Undetermined coefficients for systems
For constant forcing \(\mathbf{g} = \mathbf{b}\), try a constant \(\mathbf{x}_p = \mathbf{a}\): then \(A\mathbf{a} + \mathbf{b} = 0\), so \(\mathbf{a} = -A^{-1}\mathbf{b}\) when \(A\) is invertible. This is the system's equilibrium. For exponential or trigonometric forcing, guess vectors of the same form.
The matrix exponential
The solution of \(\mathbf{x}' = A\mathbf{x}\), \(\mathbf{x}(0) = \mathbf{x}_0\), can be written \(\mathbf{x} = e^{At}\mathbf{x}_0\), where \(e^{At} = I + At + \frac{(At)^2}{2!} + \cdots\). If \(A = PDP^{-1}\) is diagonalisable, \(e^{At} = Pe^{Dt}P^{-1}\), with \(e^{Dt}\) diagonal. It is the system version of \(e^{at}\).
e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!}Variation of parameters
For general forcing, \(\mathbf{x}(t) = e^{At}\mathbf{x}_0 + \int_0^t e^{A(t-s)}\mathbf{g}(s)\,ds\). This formula, the response to the initial state plus the accumulated response to the input, is the foundation of modern control theory.
Application: two connected tanks
Tank 1 and tank 2 each hold 100 L and exchange brine. If salt amounts satisfy \(x_1' = -0.04x_1 + 0.02x_2\) and \(x_2' = 0.04x_1 - 0.04x_2\), the eigenvalues are about \(-0.012\) and \(-0.068\), both negative, so salt levels settle. The slow eigenvalue determines how long mixing takes.
Application: coupled springs
Two masses connected by three springs obey \(m\mathbf{x}'' = K\mathbf{x}\), a second-order system. Its normal modes are in-phase motion (both masses moving together) and out-of-phase motion, each with its own frequency. Any motion is a combination of these modes; earthquake engineering uses the same idea for buildings.
Laplace transforms for systems
Transforming each equation turns a system of ODEs into a system of linear algebraic equations in \(X_1(s), X_2(s), \ldots\), solvable by elimination or matrices, then inverted. This approach handles switched and impulsive inputs neatly.
Common mistakes
Forgetting the particular solution when the system is forced. Treating the matrix exponential as entrywise exponentials, which is wrong unless \(A\) is diagonal. Ignoring the slow eigenvalue when estimating settling time.
Key takeaways
Forced systems combine homogeneous and particular solutions; constant forcing gives an equilibrium \(-A^{-1}\mathbf{b}\). The matrix exponential \(e^{At}\) solves linear systems and leads to the variation-of-parameters formula. Coupled tanks and springs illustrate eigenvalues as decay rates and normal-mode frequencies.
Flashcard
Nonhomogeneous linear system
Check your understanding
For x' = Ax + b with A invertible and constant b, the equilibrium is: