# How to (properly) use the solution of a simulation in ModelingToolkit
**URL:** https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686
**Category:** Modelling & Simulations
**Created:** [April 20, 2021, 6:39pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686 "2021-04-20T18:39:28Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@CodeLenz](https://discourse.julialang.org/u/CodeLenz)
#### Post date: [April 20, 2021, 6:39pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686/1 "2021-04-20T18:39:28Z")
Hi there.
I am starting to use the amazing ModellingToolkit, but I am afraid I am missing something important. So I apologize if my questions is silly.
Suppose I have a model consisting of two differential equations, two dependent variables x1 and x2 and an independent variable t. For example:
```julia
@parameters t a1 a2;
@variables x1(t) x2(t);
D = Differential(t);
D2 = Differential(t)^2;
f(t) = 0.1*sin(2*pi*t);
T1 = a1*f(t)
T2 = a2*f(t)
eq = [ D2(x1) ~ T1 - D(x2),
D2(x2) ~ T2 - D(x1) ];
sys = ODESystem(eq);
sys = ode_order_lowering(sys);
tspan = (0.0,10.0);
p =[a1 => 1.0,
a2 => 2.0 ];
u0 =[ D(x1) => 0.0,
x1 => 0.0,
D(x2) => 0.0,
x2 => 0.0,
];
prob = ODEProblem(sys,u0,tspan,p,jac=true)
sol = solve(prob);
```
I can plot the solution x1(t) even without knowing the position of this variable in `pos(t)`, since there is a Plot recipe
```julia
plot(sol,vars=(x1));
```
My question is. Suppose I want to make some mathematical operations on `x1(t)`. For example, evaluate the integral of `(x1(t)/(a+b))^2` over the time span.
How can I “recover” x1(t) from sol and use it along with the symbolic operations? For example,
```julia
expr = substitute( (x1/(a+b))^2, p)
```
will substitute the values of a and b, as expected. But if I try to use it with QuadGK, for example, it will give me an error, since x1(t) is not linked to sol(t).
I guess I am missing something important about it and I really want to understand how to properly use the solution provided by DifferentialEquations here.
Thank you.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [April 20, 2021, 7:38pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686/2 "2021-04-20T19:38:41Z")
`sol[x1]`?
---
### Author:  [@CodeLenz](https://discourse.julialang.org/u/CodeLenz)
#### Post date: [April 20, 2021, 8:08pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686/3 "2021-04-20T20:08:26Z")
Thank you @ChrisRackauckas
Is it possible to interpolate, like in sol(t)? I would like to call something like `sol(t)[x1]` or `sol[x1](t)..`
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [April 20, 2021, 8:15pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686/4 "2021-04-20T20:15:05Z")
Not yet. It will be `sol(t;idxs=x1)` when it’s ready.
---
### Author:  [@CodeLenz](https://discourse.julialang.org/u/CodeLenz)
#### Post date: [April 20, 2021, 8:16pm UTC](https://discourse.julialang.org/t/how-to-properly-use-the-solution-of-a-simulation-in-modelingtoolkit/59686/5 "2021-04-20T20:16:02Z")
OK.
Thank you very much!