5.1 Systems of Differential Equations

A single differential equation determines one unknown function. A system determines several at once, coupled so that none can be found without the others — two populations that prey on one another, two circuits that share a component. A first-order linear system in two unknown functions \(x(t)\) and \(y(t)\) has the form \[x' = a_{11}(t)x + a_{12}(t)y + f_1(t),\qquad y' = a_{21}(t)x + a_{22}(t)y + f_2(t),\] and its general solution contains two arbitrary constants, fixed by initial values \(x(t_0)\) and \(y(t_0)\).

The subsections below give the standard approaches. Elimination removes the unknowns one at a time until a single equation in a single unknown remains, which is then solved by the methods already available; the remaining unknowns follow by substitution. The Laplace transform turns the system into algebraic equations for the transforms. The operator method writes differentiation as \(D\) and treats the system as simultaneous algebraic equations, which is quicker once the notation is familiar and is the more reliable when the system is larger than a pair.

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