Systems of Differential Equations

A rigorous introduction to first-order systems, matrix exponentials, eigenmodes, and conversion from higher-order equations.

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A phase plane shows how two state variables change together without using time as an axis. Small arrows indicate the local direction of change, two trajectories follow them, and an equilibrium point is marked where both derivatives vanish.

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.

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 $A$.