Given an ODE, gensys returns the determining PDE system related to either its symmetries or its integrating factors, depending on the second argument. This command is typically used together with other commands to determine symmetries and integrating factors for ODEs. gensys first calls odepde to get the determining PDE and then splits it -- when possible -- by taking coefficients with respect to the derivatives of the dependent variable.

Concerning the optional second argument, gensys works like odepde; this second argument may be either an explicit form for the symmetry or the integrating factor to be taken into account at the time of calculating the determining PDE.

This function is part of the DEtools package, and so it can be used in the form gensys(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[gensys](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

$\mathrm{declare}\left(y\left(x\right)\,\mathrm{prime}=x\right)$

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=-\frac{\exp \left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)x{y\left(x\right)}^2+{y\left(x\right)}^2-\mathrm{diff}\left(y\left(x\right)\,x\right)x}{x{y\left(x\right)}^2\exp \left(x\right)}$

$\mathrm{ODE}≔\mathrm{y\text{'}\text{'}}=-\frac{{\ⅇ}^x\mathrm{y\text{'}}x{y}^2+y^2-\mathrm{y\text{'}}x}{x{y}^2{\ⅇ}^x}$

$\mathrm{gensys}\left(\mathrm{ODE}\,\mathrm{\_\μ}=\mathrm{\mu }\left(x\,y\left(x\right)\right)\right)$

${\mathrm{\mu }}_y{\ⅇ}^{x}x,{\mathrm{\mu }}_{y,y}{\ⅇ}^{x}x,{\mathrm{\mu }}_{x,y}{\ⅇ}^{x}x,-\frac{\mathrm{\mu }\left(x\,y\right)x}{y^2}-\frac{\left({\ⅇ}^x{y}^2-1\right){\mathrm{\mu }}_{x}x}{y^2}+{\mathrm{\mu }}_{x,x}{\ⅇ}^{x}x+{\mathrm{\mu }}_y$

$\mathrm{nops}\left(\left[\right]\right)$

$4$

$\mathrm{casesplit}\left(\left[\right]\right)$

$\left[{\mathrm{\mu }}_x=\mathrm{\mu }\left(x\,y\right)\,{\mathrm{\mu }}_y=0\right]\mathbf{where}\left[\right]$

$\mathrm{\mu }=\exp \left(x\right)$

$\mathrm{\mu }={\ⅇ}^x$

$\mathrm{mutest}\left(\exp \left(x\right)\,\mathrm{ODE}\right)$

$0$

$\mathrm{firint}\left(\exp \left(x\right)\mathrm{ODE}\right)$

$\frac{1}{y}+{\ⅇ}^x\mathrm{y\text{'}}+\ln \left(x\right)+\mathrm{\_C1}=0$

$\mathrm{gensys}\left(\mathrm{ODE}\,\mathrm{\_\μ}=\mathrm{\mu }\left(y\,\mathrm{\_y1}\right)\right)$

$-{\mathrm{\mu }}_{\mathrm{\_y1},\mathrm{\_y1}}{\ⅇ}^x\mathrm{\_y1}{x}^2{y}^3+{\mathrm{\mu }}_{\mathrm{\_y1},y}{\ⅇ}^x\mathrm{\_y1}{x}^2{y}^3-2{\mathrm{\mu }}_{\mathrm{\_y1}}{\ⅇ}^x{x}^2{y}^3+2{\mathrm{\mu }}_y{\ⅇ}^x{x}^2{y}^3+{\mathrm{\mu }}_{\mathrm{\_y1},\mathrm{\_y1}}\mathrm{\_y1}{x}^{2}y-{\mathrm{\mu }}_{\mathrm{\_y1},\mathrm{\_y1}}x{y}^3+2{\mathrm{\mu }}_{\mathrm{\_y1}}{x}^{2}y,-{\mathrm{\mu }}_{\mathrm{\_y1},y}{\ⅇ}^x{\mathrm{\_y1}}^2{x}^2{y}^3+{\mathrm{\_y1}}^2{\mathrm{\mu }}_{y,y}{\ⅇ}^x{x}^2{y}^3+{\mathrm{\mu }}_{\mathrm{\_y1},y}{\mathrm{\_y1}}^2{x}^{2}y-{\mathrm{\mu }}_{\mathrm{\_y1},y}\mathrm{\_y1}x{y}^3-2{\mathrm{\mu }}_{\mathrm{\_y1}}{\mathrm{\_y1}}^2{x}^2-{\mathrm{\mu }}_{\mathrm{\_y1}}\mathrm{\_y1}{x}^{2}y+{\mathrm{\mu }}_{\mathrm{\_y1}}x{y}^3+{\mathrm{\mu }}_{y}x{y}^3+{\mathrm{\mu }}_{\mathrm{\_y1}}{y}^3-\mathrm{\mu }\left(y\,\mathrm{\_y1}\right)y{x}^2$

$\mathrm{nops}\left(\left[\right]\right)$

$2$

$\mathrm{casesplit}\left(\left[\right]\right)$

$\left[\mathrm{\mu }\left(y\,\mathrm{\_y1}\right)=0\right]\mathbf{where}\left[\right]$

$\mathrm{gensys}\left(\mathrm{ODE}\,\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)$

${\mathrm{\xi }}_{y,y}{\ⅇ}^x{x}^2{y}^3,2{\ⅇ}^x{\mathrm{\xi }}_y{x}^2{y}^3+{\mathrm{\eta }}_{y,y}{\ⅇ}^x{x}^2{y}^3-2{\mathrm{\xi }}_{x,y}{\ⅇ}^x{x}^2{y}^3-2{\mathrm{\xi }}_y{x}^{2}y,{\ⅇ}^x{\mathrm{\xi }}_x{x}^2{y}^3+2{\mathrm{\eta }}_{x,y}{\ⅇ}^x{x}^2{y}^3-{\mathrm{\xi }}_{x,x}{\ⅇ}^x{x}^2{y}^3+3{\mathrm{\xi }}_{y}x{y}^3-{\mathrm{\xi }}_x{x}^{2}y+\mathrm{\xi }\left(x\,y\right)x^{2}y+2\mathrm{\eta }\left(x\,y\right)x^2,{\ⅇ}^x{\mathrm{\eta }}_x{x}^2{y}^3+{\mathrm{\eta }}_{x,x}{\ⅇ}^x{x}^2{y}^3+2{\mathrm{\xi }}_{x}x{y}^3-{\mathrm{\eta }}_{y}x{y}^3-\mathrm{\xi }\left(x\,y\right)x{y}^3-\mathrm{\xi }\left(x\,y\right)y^3-{\mathrm{\eta }}_x{x}^{2}y$

$\mathrm{nops}\left(\left[\right]\right)$

$4$

$\mathrm{casesplit}\left(\left[\right]\right)$

$\left[\mathrm{\eta }\left(x\,y\right)=0\,\mathrm{\xi }\left(x\,y\right)=0\right]\mathbf{where}\left[\right]$