ODE with a translation

Question

Is it possible to say something about the equation $$\dot{y}(t)=a(t)y(t)+b(t)y(t-T)$$ with $T>0$ a real constant?

Can we find a solution, even numerically? I don't have any clue about it.

Answer 1

Expanding upon A rural readers comment. A delay differential equation is a differential equation where the argument is delayed. This can be accounted for by numerically implementing a history function $h(p, t)$ which uses interpolations throughout the solution's history to form a continuous extension of the solver's past.

In the following I have written a solution in Julia for some arbitrary functions $a(t)$ and $b(t)$.

using DifferentialEquations
using Plots

function a(t)
    return 0.2 * t
end

function b(t)
    return 0.1 * t^2
end

function DDE(du, u, h, p, t)
    hist = h(p, t - T)[1]
    du[1] = a(t) * u[1] + b(t) * hist
end

h(p, t) = ones(1)

T = 500
lags = [T]

p = (a, b)  
tspan = (0.0, 5.0)
u0 = [2.0]

prob = DDEProblem(DDE, u0, h, tspan, p; constant_lags = lags)

# Solve the DDE
alg = MethodOfSteps(BS3())
sol = solve(prob, alg)

plot(sol)

You can change the initial conditions and the solver, for instance Rosenbrock23 or AutoVern7.

You can read more on Delay Differential Equations here