dx/dt=-x+y; dy/dt=-t*y+3x
x(0)=0; y(0)=1; t(0)=0; note t is a time varying coefficient

I tried the subsequent code but I didn’t think it calculate it correctly or maybe it’s correct but I’m unware??? My question is if there is a way to account for the time dependent parameter in the ODE? Thanks a lot.

using DifferentialEquations
using Plots
# Define the differential equations
function system!(du, u, p, t)
x, y = u
dxdt = -x + y
dydt = -t*y + 3*x
du[1] = dxdt
du[2] = dydt
end
# Initial conditions
u0 = [0.0, 1.0] # [x(0), y(0)]
# Time span
tspan = (0.0, 1.0)
# Create an ODE problem
prob = ODEProblem(system!, u0, tspan)
# Solve the ODE problem
sol = solve(prob)
# Print the solution
println(sol)
plot(sol, title="Solution of the differential equations",
xlabel="t", ylabel="Values", label=["x(t)" "y(t)"])

I don’t know the details, but I think you can try using ModelingToolkit.jl. Based on your requirements and the docs, I think the following code should meet your requirements:

beginbegin
# for modeling
using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
# for solve
using DifferentialEquations
# for plot
using CairoMakie
# define model
@mtkmodel TimeModel begin
# variables and initial values
@variables begin
x(t) = 0.0
y(t) = 1.0
end
# equations
@equations begin
# dx/dt = -x + y
D(x) ~ -x + y
# dy/dt = -t * y + 3x
D(y) ~ -t * y + 3x
end
end
# build model
@mtkbuild timemodel = TimeModel()
# solve for t ∈ [0.0, 1.0]
odeproblem = ODEProblem(timemodel, [], (0.0, 1.0), [])
solution = solve(odeproblem)
# extract values
t_table = solution.t
x_table = [solution.u[i][1] for i = 1:length(solution.u)]
y_table = [solution.u[i][2] for i = 1:length(solution.u)]
# plot solution
solution_figure = Figure()
solution_axis = Axis(solution_figure[1, 1];
aspect = 1.0,
xticks = 0.0:0.2:1.0,
yticks = 0.0:0.2:1.6,
title = "Solution of the differential equations",
titlesize = 16)
lines!(solution_axis, t_table, x_table; label = L"x\,{(t\,)}")
lines!(solution_axis, t_table, y_table; label = L"y\,{(t\,)}")
axislegend(solution_axis; position = :rb)
save("ode_solution.png", solution_figure)
solution_figure
end