Hi,

I have an ODE problem to which I need to apply delay . But trying to solve as per the documentation of delay differential equations doesn’t seem to work for me…Can someone please lend a hand?

Here is the ODE Problem I have, that induces a step response in variable u1, and correspondingly have changes in x1.

```
@parameters t, c0
D= Differential(t)
@variables x1(t), u1(t)
# input varying- u1
value_vector=[88.05,98.5]
fun(t) = t >= 400 ? value_vector[end] : value_vector[1]
@register fun(t)
eqs = [
u1 ~ fun(t),
D(x1) ~ (-x1+u1)/c0
]
@named sys = ODESystem(eqs)
simplified_sys=structural_simplify(sys)
x0=[ x1 => 88.05]
p=[c0=> 10.0 ]
prob = ODEProblem(simplified_sys,x0,(0.0,1000.0),p)
sol = solve(prob)
plot(sol)
```

Now how can I introduce a dead time in my differential equation, such that the response of x1 shall be delayed for 100 seconds after the step change I have induced in u1 after 400 sec as above? Mathematically I guess it would be something like,

D(x1) = (-x1 + u1(t - tau)) / c0 , where deadtime tau= 100

Can this be done with delay differential equations setup in DifferentialEquations.jl? Any help is highly appreciated and thanks in advance…

Here are my package specifications:

Julia Version 1.8.5

[a134a8b2] BlackBoxOptim v0.6.2

[336ed68f] CSV v0.10.10

[a93c6f00] DataFrames v1.5.0

[bcd4f6db] DelayDiffEq v5.42.0

[1130ab10] DiffEqParamEstim v2.0.1

[0c46a032] DifferentialEquations v7.7.0

[f6369f11] ForwardDiff v0.10.35

[961ee093] ModelingToolkit v8.55.0

[7f7a1694] Optimization v3.14.0

[3e6eede4] OptimizationBBO v0.1.4

[36348300] OptimizationOptimJL v0.1.8

[91a5bcdd] Plots v1.38.10

[731186ca] RecursiveArrayTools v2.38.3

[24249f21] SymPy v1.1.9