Recovering non-linear 1.order ODE from its solution

Question

Let's assume we are given a non-linear first order ODE in $n$-dimensions in $\mathbb{R}^n$ or some open subset:

$\begin{matrix} \dot { { x }_{ 1 } } & = & f_{ 1 }(x_{ 1 },...,x_{ n }) \\ \vdots & & \\ \dot { { x }_{ n } } & = & f_{ n }(x_{ 1 },...,x_{ n }) \end{matrix}$

Further we assume that we know a solution $\gamma(x_0,t):\mathbb{R}^n \rightarrow \mathbb{R}^n$ for an initial condition $x_0$.

Is there a way to reconstruct in general the form of the ODE, i.e. the functions $f_i,\ i=1,...,n$ ?

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I am aware of a solution for linear ODEs. I also assume there is no unique solution to the problem. If this is the case I am interested in finding all possible ODEs which could belong to the given solution.

Answer 1

Well, in a local set up, you need to know the general solution, that is, $\gamma(x_0,t)$ for any $x_0=(c_1,\ldots,c_n)\in \mathbb{R}^n$. Alternatively, you can have $\gamma$ depending on another set of "equivalent" constants.
Suppose $$ \gamma((c_1,\ldots,c_n),t)=(x_1((c_1,\ldots,c_n),t)\ldots,x_n((c_1,\ldots,c_n),t)). $$

You only need to define $$ \varphi: ((c_1,\ldots,c_n),t)\to (t,x_1,\ldots,x_n) $$ and compute $\varphi^{-1}$. Then, you compute the pushforward of $\partial_t$: $$ A:=\varphi_*(\partial_t) $$ (observe that you need $\varphi^{-1}$ to calculate the components).

This vector field $A$ is $$ A=\partial_t+f_1 \partial_{x_1}+f_2 \partial_{x_2}+\ldots+f_n \partial_{x_n}, $$ being $f_i$ your desired functions. Observe that depending on the expressions of the solution, the system could be non-autonomous, i.e., $f_i$ can depend on $t$.

Here you have a Maple code to compute the ODE system from a solution:

restart; 
with(DifferentialGeometry); with(JetCalculus); with(Tools); with(ExteriorDifferentialSystems); with(PDEtools); with(LieAlgebras); with(LinearAlgebra); with(DEtools); with(Student[LinearAlgebra]); 

DGsetup([s, c1, c2, c3, c4], M1); 
baseformas1 := Tools:-DGinfo(M1, "FrameJetForms"); 
basecampos1 := Tools:-DGinfo(M1, "FrameJetVectors"); 

DGsetup([t, x, y, z, w], M2); 
baseformas2 := Tools:-DGinfo(M2, "FrameJetForms"); 
basecampos2 := Tools:-DGinfo(M2, "FrameJetVectors");

xsol := (c1*s+c2)/(s-c3); 
ysol := c2/(c4-s);
zsol := (c4*s+c3)/s; 
wsol := c4/s+c1;

phi := Transformation(M1, M2, [t = s, x = xsol, y = ysol, z = zsol, w = wsol]); 
invphi := InverseTransformation(phi);

A := simplify(Pushforward(phi, invphi, basecampos1[1]));

I have to say that if you have only one particular solution you can add to it enough constants to create a family of solutions as above, whenever these constants are "suficientely independent". Then your solution will be a solution of the just created ODE system.

Answer 2

I think the problem is not that simple so that we only have to look at the dimensions of the system. Take in $1D$ the solution $$\tag{1}\gamma(1,t)=\frac{1}{t}\,\,,\quad t\in(0,+\infty)\,.$$ It solves the ODE $\dot\gamma=-\gamma^2\,.$ Which we can write as $\dot\gamma=f(\gamma)$ with $f(x)=-x^2\,.$ This $f$ is an even function that is defined on the whole of $\mathbb R$ but the particular solution (1) does clearly not care when we modify $f$ on $(-\infty,0]\,.$ In other words:

• For any function $g$ on $\mathbb R$ that agrees with $f(x)=-x^2$ on $(0,+\infty)$ the function (1) is a solution of the ODE $\dot\gamma=g(\gamma)\,.$

I think the problem you are considering has to do with the range of $\gamma$ and with the connectedness of the domain of $f\,.$