The rifsimp function can be used to simplify or rework overdetermined systems of polynomially nonlinear PDEs or ODEs and inequations to a more useful form -- relative standard form. The rifsimp function does not solve PDE systems, but provides existence and uniqueness information, and can be used as a first step to their solution. As an example, inconsistent systems can be detected by rifsimp.

Basically, given an input PDE system, and a list of dependent variables or constants to solve for, rifsimp returns the simplified PDE system along with any existence conditions required for the simplified system to hold.

Detailed examples of the use of rifsimp for various systems (along with some explanation of the algorithm) can be found in rifsimp[overview].

Other options are sometimes required along with the specification of the system and its solving variables. For common options, please see rifsimp[options], and for more advanced use, please see rifsimp[adv_options].

For a description of all possible output configurations, see rifsimp[output].

$\mathrm{sys1}≔\left[x{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2{\mathrm{diff}\left(y\left(x\right)\,x\,x\right)}^2-2x\mathrm{diff}\left(y\left(x\right)\,x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+x{y\left(x\right)}^2{\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)}^2-y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2=0\,-\mathrm{diff}\left(y\left(x\right)\,x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+2{y\left(x\right)}^2{\mathrm{diff}\left(y\left(x\right)\,x\,x\right)}^2-4y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right){\mathrm{diff}\left(y\left(x\right)\,x\right)}^2+2{\mathrm{diff}\left(y\left(x\right)\,x\right)}^4=0\right]$

$\mathrm{sys1}≔\left[x{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}^2-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)y\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+x{y\left(x\right)}^2{\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}^2-y\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2=0\,-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+y\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+2{y\left(x\right)}^2{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}^2-4y\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+2{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^4=0\right]$

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

$\mathrm{ans1}≔\mathrm{rifsimp}\left(\mathrm{sys1}\right)$

$\mathrm{ans1}≔table\left(\left[\mathrm{Case}=\left[\left[y\left(x\right)\ne 0\,\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right]\right]\,\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}\right]\,\mathrm{Pivots}=\left[y\left(x\right)\ne 0\right]\right]\right)$

$\mathrm{dsolve}\left(\mathrm{ans1}\left[\mathrm{Solved}\right]\,\left\{y\left(x\right)\right\}\,\mathrm{explicit}\right)$

$\left\{y\left(x\right)={\ⅇ}^{\mathrm{c\_\_1}x}\mathrm{c\_\_2}\right\}$

$\mathrm{id1}≔\mathrm{initialdata}\left(\mathrm{ans1}\left[\mathrm{Solved}\right]\right)$

$\mathrm{id1}≔table\left(\left[\mathrm{Finite}=\left[y\left(x_0\right)=\mathrm{c\_\_1}\,\mathrm{D}\left(y\right)\left(x_0\right)=\mathrm{c\_\_2}\right]\,\mathrm{Infinite}=\left[\right]\right]\right)$

$\mathrm{dsolve}\left(\mathrm{convert}\left(\mathrm{ans1}\left[\mathrm{Solved}\right]\,\mathrm{set}\right)\mathbf{union}\left\{y\left(0\right)=1\,\mathrm{D}\left(y\right)\left(0\right)=2\right\}\,\left\{y\left(x\right)\right\}\,\mathrm{numeric}\,\mathrm{output}=\mathrm{Array}\left(\left[0\,0.25\,0.5\,0.75\,1.0\right]\right)\right)$

$\left[\begin{array}{c}\left[\begin{array}{ccc}x& y\left(x\right)& \frac{\ⅆ}{\ⅆx}y\left(x\right)\end{array}\right]\\ \left[\begin{array}{ccc}0.& 1.& 2.\\ 0.250000000000000& 1.64872125401162& 3.29744250802324\\ 0.500000000000000& 2.71828135845628& 5.43656271691257\\ 0.750000000000000& 4.48168797609521& 8.96337595219041\\ 1.& 7.38905394806921& 14.7781078961384\end{array}\right]\end{array}\right]$

$\mathrm{tay\_ser}≔\mathrm{rtaylor}\left(\mathrm{ans1}\left[\mathrm{Solved}\right]\,\mathrm{order}=4\right)$

$\mathrm{tay\_ser}≔\left[y\left(x\right)=y\left(x_0\right)+\mathrm{D}\left(y\right)\left(x_0\right)\left(x-x_0\right)+\frac{{\mathrm{D}\left(y\right)\left(x_0\right)}^2{\left(x-x_0\right)}^2}{2y\left(x_0\right)}+\frac{{\mathrm{D}\left(y\right)\left(x_0\right)}^3{\left(x-x_0\right)}^3}{6{y\left(x_0\right)}^2}+\frac{{\mathrm{D}\left(y\right)\left(x_0\right)}^4{\left(x-x_0\right)}^4}{24{y\left(x_0\right)}^3}\right]$

$\mathrm{tay\_ser}≔\mathrm{rtaylor}\left(\mathrm{ans1}\left[\mathrm{Solved}\right]\,\mathrm{id1}\,\mathrm{order}=4\right)$

$\mathrm{tay\_ser}≔\left[y\left(x\right)=\mathrm{c\_\_1}+\mathrm{c\_\_2}\left(x-x_0\right)+\frac{{\mathrm{c\_\_2}}^2{\left(x-x_0\right)}^2}{2\mathrm{c\_\_1}}+\frac{{\mathrm{c\_\_2}}^3{\left(x-x_0\right)}^3}{6{\mathrm{c\_\_1}}^2}+\frac{{\mathrm{c\_\_2}}^4{\left(x-x_0\right)}^4}{24{\mathrm{c\_\_1}}^3}\right]$

$\mathrm{sys2}≔\left[{\mathrm{diff}\left(u\left(x\,y\right)\,x\right)}^2+\mathrm{diff}\left(u\left(x\,y\right)\,y\,y\right)=0\,{u\left(x\,y\right)}^3-\mathrm{diff}\left(u\left(x\,y\right)\,x\right)+x=0\right]\:$

$\mathrm{rifsimp}\left(\mathrm{sys2}\right)$

$table\left(\left[\mathrm{status}=''system\; is\; inconsistent''\right]\right)$

$\mathrm{ConstrainedSys}≔\left[m\ne 0\,\mathrm{\Phi }\left(x\left(t\right)\,y\left(t\right)\right)=0\,m\mathrm{diff}\left(x\left(t\right)\,t\,t\right)=-\mathrm{\lambda }\left(t\right)\mathrm{D}\left[1\right]\left(\mathrm{\Phi }\right)\left(x\left(t\right)\,y\left(t\right)\right)\,m\mathrm{diff}\left(y\left(t\right)\,t\,t\right)=-\mathrm{\lambda }\left(t\right)\mathrm{D}\left[2\right]\left(\mathrm{\Phi }\right)\left(x\left(t\right)\,y\left(t\right)\right)-mg\right]$

$\mathrm{ConstrainedSys}≔\left[m\ne 0\,\mathrm{\Phi }\left(x\left(t\right)\,y\left(t\right)\right)=0\,m\left(\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right)=-\mathrm{\lambda }\left(t\right){\mathrm{D}}_1\left(\mathrm{\Phi }\right)\left(x\left(t\right)\,y\left(t\right)\right)\,m\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)=-\mathrm{\lambda }\left(t\right){\mathrm{D}}_2\left(\mathrm{\Phi }\right)\left(x\left(t\right)\,y\left(t\right)\right)-mg\right]$

$\mathrm{\Phi }≔\left(x\,y\right)\mapsto 0$

$\mathrm{\Phi }≔\left(x\,y\right)\mapsto 0$

$\mathrm{\Phi }≔\left(x\,y\right)\mapsto x^2+y^2-1$

$\mathrm{\Phi }≔\left(x\,y\right)\mapsto x^2+y^2-1$

$\mathrm{ConstrainedSys}$

$\left[m\ne 0\,{x\left(t\right)}^2+{y\left(t\right)}^2-1=0\,m\left(\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right)=-2\mathrm{\lambda }\left(t\right)x\left(t\right)\,m\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)=-2\mathrm{\lambda }\left(t\right)y\left(t\right)-mg\right]$

$\mathrm{\Φ2}≔\left(x\,y\right)\mapsto x^4+y^4-1$

$\mathrm{\Φ2}≔\left(x\,y\right)\mapsto x^4+y^4-1$

$\mathrm{SimpSys}≔\mathrm{rifsimp}\left(\mathrm{ConstrainedSys}\right)$

$\mathrm{SimpSys}≔table\left(\left[\mathrm{Constraint}=\left[mg{y\left(t\right)}^3+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^{2}m-mgy\left(t\right)+2\mathrm{\lambda }\left(t\right){y\left(t\right)}^2-2\mathrm{\lambda }\left(t\right)=0\,{x\left(t\right)}^2+{y\left(t\right)}^2-1=0\right]\,\mathrm{Case}=\left[\left[x\left(t\right)\ne 0\,\frac{\ⅆ}{\ⅆt}x\left(t\right)\right]\,\left[\left(y\left(t\right)-1\right)\left(y\left(t\right)+1\right)\ne 0\,\frac{\ⅆ}{\ⅆt}\mathrm{\lambda }\left(t\right)\right]\right]\,\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)=-\frac{mg+2\mathrm{\lambda }\left(t\right)y\left(t\right)}{m}\,\frac{\ⅆ}{\ⅆt}\mathrm{\lambda }\left(t\right)=-\frac{3mg\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}{2}\,\frac{\ⅆ}{\ⅆt}x\left(t\right)=-\frac{y\left(t\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}{x\left(t\right)}\right]\,\mathrm{Pivots}=\left[m\ne 0\,x\left(t\right)\ne 0\,y\left(t\right)-1\ne 0\,y\left(t\right)+1\ne 0\right]\right]\right)$

$\mathrm{initialdata}\left(\mathrm{SimpSys}\right)$

$table\left(\left[\mathrm{Constraint}=\left[{\mathrm{\_C3}}^3\mathrm{\_C5}\mathrm{\_C6}+2{\mathrm{\_C3}}^2\mathrm{c\_\_1}-\mathrm{\_C3}\mathrm{\_C5}\mathrm{\_C6}+{\mathrm{\_C4}}^2\mathrm{\_C6}-2\mathrm{c\_\_1}=0\,{\mathrm{\_C3}}^2+{\mathrm{c\_\_2}}^2-1=0\right]\,\mathrm{Finite}=\left[\mathrm{\lambda }\left(t_0\right)=\mathrm{c\_\_1}\,x\left(t_0\right)=\mathrm{c\_\_2}\,y\left(t_0\right)=\mathrm{\_C3}\,\mathrm{D}\left(y\right)\left(t_0\right)=\mathrm{\_C4}\,g=\mathrm{\_C5}\,m=\mathrm{\_C6}\right]\,\mathrm{Infinite}=\left[\right]\,\mathrm{Pivots}=\left[\mathrm{\_C6}\ne 0\,\mathrm{c\_\_2}\ne 0\,\mathrm{\_C3}-1\ne 0\,\mathrm{\_C3}+1\ne 0\right]\right]\right)$

$\mathrm{ODE}≔2x^{2}y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x^2\left({\mathrm{diff}\left(y\left(x\right)\,x\right)}^2+1\right)+{y\left(x\right)}^2$

$\mathrm{ODE}≔2x^{2}y\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x^2\left({\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+1\right)+{y\left(x\right)}^2$

$\mathrm{sys}≔\left\{\mathrm{coeffs}\left(\mathrm{numer}\left(\mathrm{DEtools}\left[\mathrm{odepde}\right]\left(\mathrm{ODE}\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\left(x\,y\right)\,y\left(x\right)\right)\right)\,\mathrm{\_y1}\right)\right\}$

$\mathrm{sys}≔\left\{-2\left(\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)\right)x^3{y}^2-\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)x^{3}y\,2\left(\frac{{\partial }^2}{\partial y^2}\mathrm{\eta }\left(x\,y\right)\right)x^3{y}^2-4\left(\frac{{\partial }^2}{\partial x\partial y}\mathrm{\xi }\left(x\,y\right)\right)x^3{y}^2-\left(\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)x^{3}y+\mathrm{\eta }\left(x\,y\right)x^3\,4\left(\frac{{\partial }^2}{\partial x\partial y}\mathrm{\eta }\left(x\,y\right)\right)x^3{y}^2-2\left(\frac{{\partial }^2}{\partial x^2}\mathrm{\xi }\left(x\,y\right)\right)x^3{y}^2-3\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)x^{3}y+3\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)x{y}^3-2\left(\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)\right)x^{3}y\,2\left(\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\right)\right)x^3{y}^2-2\left(\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\right)x^{3}y+2\left(\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\right)x{y}^3+\left(\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)x^{3}y-\left(\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)x{y}^3+\mathrm{\eta }\left(x\,y\right)x^3+\mathrm{\eta }\left(x\,y\right)x{y}^2-2\mathrm{\xi }\left(x\,y\right)y^3\right\}$

$\mathrm{rifsys}≔\mathrm{rifsimp}\left(\mathrm{sys}\right)$

$\mathrm{rifsys}≔table\left(\left[\mathrm{Solved}=\left[\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\right)=\frac{-\mathrm{\eta }\left(x\,y\right)x+\mathrm{\xi }\left(x\,y\right)y}{x^3}\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=\frac{\mathrm{\eta }\left(x\,y\right)}{y}\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=\frac{\mathrm{\eta }\left(x\,y\right)}{y}\,\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)=0\right]\right]\right)$

$\mathrm{pdsolve}\left(\mathrm{rifsys}\left['\mathrm{Solved}'\right]\right)$

$\left\{\mathrm{\eta }\left(x\,y\right)=\left(\mathrm{c\_\_3}{\ln \left(x\right)}^2+\mathrm{c\_\_2}\ln \left(x\right)+\mathrm{c\_\_1}\right)y\,\mathrm{\xi }\left(x\,y\right)=x\left(\mathrm{c\_\_3}{\ln \left(x\right)}^2+\left(\mathrm{c\_\_2}-2\mathrm{c\_\_3}\right)\ln \left(x\right)+\mathrm{c\_\_1}-\mathrm{c\_\_2}+2\mathrm{c\_\_3}\right)\right\}$