# Finding Parameters for Boundary-Value Problem using DifferentialEquations.jl

**URL:** <https://discourse.julialang.org/t/finding-parameters-for-boundary-value-problem-using-differentialequations-jl/129386>\
**Category:** Numerics\
**Tags:** differentialequation\
**Created:** [May 27, 2025, 3:02pm UTC](https://discourse.julialang.org/t/finding-parameters-for-boundary-value-problem-using-differentialequations-jl/129386 "2025-05-27T15:02:34Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![parf](https://avatars.discourse-cdn.com/v4/letter/p/779978/32.png) [@parf](https://discourse.julialang.org/u/parf)\
**Post date:** [May 27, 2025, 3:02pm UTC](https://discourse.julialang.org/t/finding-parameters-for-boundary-value-problem-using-differentialequations-jl/129386/1 "2025-05-27T15:02:34Z")

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Hello everyone!

I have a system of equations, the solution of which defines a trajectory on a plane:

```julia
dx/dt = Vx(x,y,t,p),
dy/dt = Vy(x,y,t,p),

```

where `p` is some unknown parameter. I know the starting point of the trajectory, i.e. `(x[0], y[0]) = (xA, yA)`. My task is to find the value of the parameter `p` at which the trajectory will arrive at the point `(xB, yB)` in the shortest time. Also, I need to find the corresponding trajectory and travel time `T`.

I’m not sure what type this problem is and whether it can be solved with `DifferentialEquations.jl`? Note, that one can transform the variable `t' = t/T` and then the timespan of the problem becomes fixed `[0,1]`, but we have an additional unknown parameter `T` in the transformed system of equations. So, how can we find the values ​​of the parameters `(p, T)` of a system of equations if the solution is known at two fixed points?

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**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 27, 2025, 5:06pm UTC](https://discourse.julialang.org/t/finding-parameters-for-boundary-value-problem-using-differentialequations-jl/129386/2 "2025-05-27T17:06:48Z")

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Just make it into a parameter estimation problem and optimize it:

> **[Parameter Estimation of Ordinary Differential Equations · SciMLSensitivity.jl](https://docs.sciml.ai/SciMLSensitivity/stable/tutorials/parameter_estimation_ode/)**
>
> Documentation for SciMLSensitivity.jl.

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<div class="post-metadata">

**Author:** ![parf](https://avatars.discourse-cdn.com/v4/letter/p/779978/32.png) [@parf](https://discourse.julialang.org/u/parf)\
**Post date:** [May 28, 2025, 9:50am UTC](https://discourse.julialang.org/t/finding-parameters-for-boundary-value-problem-using-differentialequations-jl/129386/3 "2025-05-28T09:50:21Z")

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Thank you! I will try.