# How to construct the variational equation of an ODE without handwriting the Jacobi matrix

**URL:** <https://discourse.julialang.org/t/how-to-construct-the-variational-equation-of-an-ode-without-handwriting-the-jacobi-matrix/98735>\
**Category:** Numerics\
**Tags:** question, differentialequation\
**Created:** [May 12, 2023, 11:22am UTC](https://discourse.julialang.org/t/how-to-construct-the-variational-equation-of-an-ode-without-handwriting-the-jacobi-matrix/98735 "2023-05-12T11:22:31Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![xiaoming](https://avatars.discourse-cdn.com/v4/letter/x/a6a055/32.png) [@xiaoming](https://discourse.julialang.org/u/xiaoming)\
**Post date:** [May 12, 2023, 11:22am UTC](https://discourse.julialang.org/t/how-to-construct-the-variational-equation-of-an-ode-without-handwriting-the-jacobi-matrix/98735/1 "2023-05-12T11:22:31Z")

</div>

Hello! I am new to Julia. I want to solve the variational equation of an ODE without handwriting the Jacobi matrix. Here is an MME. Suppose that the ODE equations are the followings:

\dot{x}=y\\ \dot{y}=x-x^3-0.1y+0.1\cos(t)

Set f(x,y,t)=(y,x-x^3-0.1y+0.1\cos(t))^T. Given a initial condition (x\_0,y\_0) at t=0, suppose the solution is (x(t),y(t)) for t\in[0,2\pi]. Now I want to solve the following matrix ODE:

\dot{X}=d\_{(x,y)}f(x(t),y(t),t)X

with initial condition X(0)=id. I can solve the above matrix ODE by using the following codes:

```julia
using OrdinaryDiffEq, StaticArrays

function vectorfield(u, p, t)
    SA[u[2], u[1]-u[1]^3-0.1*u[2]+0.1*cos(t)] # vector field of the system
end

function timemap(u0)
    tspan = (0.0, 2 * pi)
    prob = ODEProblem(vectorfield, u0, tspan)
    sol = solve(prob, Vern7())
    function dv(X, p, t)
        SA[0.0 0.1; 1-3*(sol(t)[1])^2 -0.1] * X # variational equation in matrix form
    end
    X0 = SA[1.0 0.0; 0.0 1.0] # initial condition of variational equation
    probJ = ODEProblem(dv, X0, tspan)
    solJ = solve(probJ, Vern7(), save_everystep=false)
    (sol[end], solJ[end])
end

```

However, when the ODE has a large dimension, hand calculation of the Jacobi matrix is not a good choice. Can anyone show me how to solve the matrix ODE like the above example without hand calculation of the Jacobi matrix?

---

<div class="post-metadata">

**Author:** ![gdalle](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gdalle/32/27854_2.png) [@gdalle](https://discourse.julialang.org/u/gdalle)\
**Post date:** [May 12, 2023, 2:50pm UTC](https://discourse.julialang.org/t/how-to-construct-the-variational-equation-of-an-ode-without-handwriting-the-jacobi-matrix/98735/2 "2023-05-12T14:50:38Z")

</div>

What you’re looking for seems to be [automatic (or algorithmic) differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation). How large is “large dimension”? Depending on the answer, there can be several packages for computing the Jacobian: see the list at [https://juliadiff.org/](https://juliadiff.org/)