First-order homogeneous linear discrete delay differential equation with constant coefficients

Definition

A first-order linear delay differential equation with constant coefficients is a particular type of delay differential equation: a first-order delay differential equation that is linear and where the coefficients are all constants. Explicitly, if we denote the independent variable by $t$ and the dependent variable by $x$ , then there are constants $A_{0},A_{1},\dots ,A_{m}$ and positive constants $\tau _{1},\tau _{2},\dots ,\tau _{m}$ such that the delay differential equation has the form:

${\frac {d}{dt}}(x(t))=A_{0}x(t)+\sum _{j=1}^{m}A_{j}x(t-\tau _{j})$

Procedure for finding smooth solutions

Although there may well exist many non-smooth solutions, the globally analytic solutions are quite limited. Explicitly, we construct first a characteristic equation as an equation in $z$ :

$z=A_{0}+\sum _{j=1}^{m}A_{j}e^{-z\tau _{j}}$

If $\alpha$ is a solution of this equation, it is termed a characteristic value or eigenvalue and the function $x(t)=e^{\alpha t}$ is a solution of the original linear delay differential equation. Since the delay differential equation is linear, any linear combination of such functions (for different characteristic values) also gives a solution function.

Note that, in general, we need to look for both real and complex solutions to this equation. Complex solutions may still give real function by the usual trick of taking the real and imaginary parts. Explicitly, if $a\pm ib$ is a conjugate pair of complex solutions to the differential equation, then the corresponding solution functions are $e^{at}\cos(bt)$ and $e^{at}\sin(bt)$ .

Note also that there is no generic way of bounding the number of solutions, and, particularly if we are looking over the complex numbers, it is often the case that there are infinitely many solutions, giving rise to an infinite-dimensional solution space of linear combinations.

The procedure generalizes to solving an arbitrary order linear delay differential equation with constant coefficients.

Examples

Consider the linear delay differential equation:

${\frac {d}{dt}}(x(t))=x(t)+4x(t-\ln 2)$

The characteristic equation is:

$z=1+4e^{-z\ln 2}$

This simplifies to:

$z=1+{\frac {4}{2^{z}}}$

We have to solve this equation for $z$ . Let's first try to find real solutions.

Finding real solutions

If $z$ is real, then we know that $2^{z}>0$ , so ${\frac {4}{2^{z}}}>0$ , so $1+{\frac {4}{2^{z}}}>1$ , so $z>1$ . But since $z>1$ , $2^{z}>2$ , so ${\frac {4}{2^{z}}}<2$ , so $z<1+2=3$ . Thus, $z\in (1,3)$ .

Further, we note that while the left side of the equation is increasing in $z$ , the right side is decreasing in $z$ , so the equation has a unique solution in $(1,3)$ . A quick inspection shows that this unique solution is $z=2$ .

Thus, we get one family of solutions:

$x(t)=Ce^{2t},C\in \mathbb {R}$

However, there may well be other complex solutions to the characteristic equation, giving rise to other solutions to the original delay differential equation.