In the jet bundle approach to the geometric theory of differential equations, a system of $k$-th order differential equations is represented by a sub-manifold $R^k$ of a jet space $J^k\left(E\right)$, where $\mathrm{\π} \:E \to M$ is a fiber bundle. A solution to the differential equation defined by the sub-manifold $R^k$is a section $s\:M \to E$whose jets $j^k\left(s\right)\left(x\right)$take values in $R^k$.

The DifferentialEquationData command creates an internal data structure which allows for the subsequent manipulation of the system of differential equations. The differential equations can be formally prolonged to higher order jet spaces using the Prolong command. The imbedding $\mathrm{\φ}\:R^k\to J^k\left(E\right)$can be constructed using the Transformation command.

The command DifferentialEquationData is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form DifferentialEquationData(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-DifferentialEquationData(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,E\,1\right)\:$

$\mathrm{DE}≔\mathrm{DifferentialEquationData}\left(\left[u\left[1,2\right]=\sin \left(u\left[\right]\right)\right]\,\left[u\left[1,2\right]\right]\right)$

$\mathrm{DE}≔\left[\left\{u_{1,2}\right\}\,\left[u_{1,2}-\sin \left(u_{\left[\right]}\right)\right]\right]$

$\mathrm{DE1}≔\mathrm{Prolong}\left(\mathrm{DE}\,1\right)$

$\mathrm{DE1}≔\left[\left\{u_{1,2}\,u_{1,1,2}\,u_{1,2,2}\right\}\,\left[u_{1,2}-\sin \left(u_{\left[\right]}\right)\,-\cos \left(u_{\left[\right]}\right)u_1+u_{1,1,2}\,-\cos \left(u_{\left[\right]}\right)u_2+u_{1,2,2}\right]\right]$

$\mathrm{\iota }≔\mathrm{Transformation}\left(\mathrm{DE1}\right)$

$\mathrm{\iota }≔\left[x=x\,y=y\,u_{\left[\right]}=u_{\left[\right]}\,u_1=u_1\,u_2=u_2\,u_{1,1}=u_{1,1}\,u_{1,2}=\sin \left(u_{\left[\right]}\right)\,u_{2,2}=u_{2,2}\,u_{1,1,1}=u_{1,1,1}\,u_{1,1,2}=\cos \left(u_{\left[\right]}\right)u_1\,u_{1,2,2}=\cos \left(u_{\left[\right]}\right)u_2\,u_{2,2,2}=u_{2,2,2}\right]$

$f≔-\cos \left(u\left[\right]\right)u\left[1\right]u\left[2\right]+{u\left[1,2\right]}^2-u\left[1,2\right]\sin \left(u\left[\right]\right)+u\left[2\right]u\left[1,1,2\right]$

$f≔-\cos \left(u_{\left[\right]}\right)u_1{u}_2+u_{1,2}^2-u_{1,2}\sin \left(u_{\left[\right]}\right)+u_2{u}_{1,1,2}$

$\mathrm{Pullback}\left(\mathrm{\iota }\,f\right)$

$0$