Systems of Differential Equations

Precise definition

A first-order system has vector form $\mathbf x'=\mathbf f(t,\mathbf x)$. A linear system is $\mathbf x'=A(t)\mathbf x+\mathbf g(t)$; for constant $A$ and no forcing, the solution is $\mathbf x(t)=e^{At}\mathbf x(0)$. If $A$ has enough independent eigenvectors, homogeneous modes are $e^{\lambda t}\mathbf v$.

Notation and mathematical language

A second-order scalar equation can become a system by setting $x_1=y$, $x_2=y'$, so $x_1'=x_2$ and $x_2'$ comes from the original equation. Initial data form a state vector. Eigenvalues control modal growth, decay, and oscillation.

Conceptual picture

A system tracks interacting state components simultaneously. Eigenvectors give invariant directions for a linear autonomous system; along one, the vector field changes only the mode's magnitude. General initial states decompose into modes when a basis of eigenvectors exists.

Conditions and key results

Constant-matrix eigenvector formulas require diagonalizability for a full eigenbasis; defective matrices need generalized eigenvectors or direct matrix exponentials. Complex modes combine into real solutions for real systems. Nonlinear systems are only locally approximated by their Jacobian.

A reliable strategy

1. Choose state variables and write every equation first order with consistent units.
2. For a constant linear homogeneous system, find eigenvalues and eigenvectors, including multiplicities.
3. Build the real general solution, use generalized vectors if needed, and apply the initial state.
4. Differentiate the vector solution and verify $\mathbf x'=A\mathbf x+\mathbf g$.

Fully worked example

Interpretation and application

Systems model compartments, coupled oscillators, chemical networks, and control states. Interaction coefficients describe the chosen model; observed correlation among state variables does not independently prove the directed causal couplings in $A$.

Common mistakes

Verification and reasonableness

• Compute $A\mathbf x$ and compare with the differentiated solution componentwise.
• Check $\mathbf x(0)$ exactly and verify trace/determinant against eigenvalue sum/product.
• Plot or inspect mode signs to confirm the phase-plane classification.

Practice

1. Convert $y''+y=0$ using $x_1=y,x_2=y'$.
2. What mode corresponds to eigenpair $(\lambda,\mathbf v)$?
3. What if a matrix lacks enough eigenvectors?

Further deduction

For a defective example $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, there is one eigenvector for the double eigenvalue 1. Direct exponentiation gives $e^{At}=e^t\begin{pmatrix}1&t\\0&1\end{pmatrix}$. The factor $t e^t$ is the system analogue of the repeated-root term in scalar constant-coefficient equations.

For a nonhomogeneous constant system, variation of constants gives $x(t)=e^{A(t-t_0)}x_0+\int_{t_0}^t e^{A(t-s)}g(s)ds$. The first term propagates the initial state; the integral accumulates every input after it is propagated for the remaining time. Differentiating the expression verifies both the forcing and initial condition, and makes the scalar integrating-factor formula a one-dimensional special case.

Related topics

• Phase Plane
• Eigenvalues
• Second-Order Linear Differential Equations