ConsistencyTest tests whether a system of equations, possibly containing inequations and involving or not differential equations, is or not consistent. A system is inconsistent when one of its equations contradicts one other one. In order to be solvable, a system must be consistent in the first place.

If DepVars is not specified, ConsistencyTest will consider all the differentiated unknown functions in PDESYS as unknown of the problems.

By default, ConsistencyTest returns true or false. If the system is consistent when the option argument outputthesystem is specified, ConsistencyTest returns three lists with equations, inequations and function replacements to represent PDESYS in differential polynomial form, where the equations and inequations are equivalent to PDESYS simplified with respect to its integrability conditions. These three lists  returned can also be computed using dpolyform, but the number of operations performed to compute it is smaller when using ConsistencyTest. If PDESYS happens to be inconsistent and outputthesystem is specified ConsistencyTest returns NULL.

To obtain output without auxiliary functions when outputthesystem is specified use in addition the optional argument no_Fn.

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

$\mathrm{declare}\left(f\left(x\,y\right)\,g\left(x\,y\right)\right)$

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

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

$\mathrm{sys}\left[1\right]≔\left[\mathrm{diff}\left(f\left(x\,y\right)\,y\,y\right)=0\,-6\mathrm{diff}\left(f\left(x\,y\right)\,y\right)y+\mathrm{diff}\left(g\left(x\,y\right)\,y\,y\right)-2\mathrm{diff}\left(f\left(x\,y\right)\,x\,y\right)=0\,-9\mathrm{diff}\left(f\left(x\,y\right)\,y\right)a{y}^2-12\mathrm{diff}\left(f\left(x\,y\right)\,y\right)a^{2}y-3\mathrm{diff}\left(f\left(x\,y\right)\,y\right)b-3\mathrm{diff}\left(f\left(x\,y\right)\,x\right)y+2\mathrm{diff}\left(g\left(x\,y\right)\,x\,y\right)-\mathrm{diff}\left(f\left(x\,y\right)\,x\,x\right)-3g\left(x\,y\right)=0\,-6g\left(x\,y\right)ay-6\mathrm{diff}\left(f\left(x\,y\right)\,x\right)a{y}^2-8\mathrm{diff}\left(f\left(x\,y\right)\,x\right)a^{2}y+3\mathrm{diff}\left(g\left(x\,y\right)\,y\right)a{y}^2+4\mathrm{diff}\left(g\left(x\,y\right)\,y\right)a^{2}y-4g\left(x\,y\right)a^2-2\mathrm{diff}\left(f\left(x\,y\right)\,x\right)b+\mathrm{diff}\left(g\left(x\,y\right)\,y\right)b-3y\mathrm{diff}\left(g\left(x\,y\right)\,x\right)+\mathrm{diff}\left(g\left(x\,y\right)\,x\,x\right)=0\right]$

${\mathrm{sys}}_1≔\left[f_{y,y}=0\,-6f_{y}y+g_{y,y}-2f_{x,y}=0\,-9f_{y}a{y}^2-12f_y{a}^{2}y-3f_{y}b-3f_{x}y+2g_{x,y}-f_{x,x}-3g=0\,-6gay-6f_{x}a{y}^2-8f_x{a}^{2}y+3g_{y}a{y}^2+4g_y{a}^{2}y-4g{a}^2-2f_{x}b+g_{y}b-3y{g}_x+g_{x,x}=0\right]$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[1\right]\right)$

$\mathrm{true}$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[1\right]\,\mathrm{outputthesystem}\right)$

$\left[g=0\,f_x=0\,f_y=0\right],\left[\right],\left[\right]$

$\mathrm{sys}\left[2\right]≔\left[\mathrm{op}\left(\mathrm{sys}\left[1\right]\right)\,\mathrm{expand}\left(\mathrm{lhs}\left(\mathrm{sys}\left[1\right]\left[1\right]\right)\mathrm{lhs}\left(\mathrm{sys}\left[1\right]\left[2\right]\right)\right)=1\right]$

${\mathrm{sys}}_2≔\left[f_{y,y}=0\,-6f_{y}y+g_{y,y}-2f_{x,y}=0\,-9f_{y}a{y}^2-12f_y{a}^{2}y-3f_{y}b-3f_{x}y+2g_{x,y}-f_{x,x}-3g=0\,-6gay-6f_{x}a{y}^2-8f_x{a}^{2}y+3g_{y}a{y}^2+4g_y{a}^{2}y-4g{a}^2-2f_{x}b+g_{y}b-3y{g}_x+g_{x,x}=0\,-6f_{y,y}{f}_{y}y+f_{y,y}{g}_{y,y}-2f_{y,y}{f}_{x,y}=1\right]$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[2\right]\right)$

$\mathrm{false}$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[2\right]\,\mathrm{outputthesystem}\right)$

Warning: System is inconsistent

$\mathrm{sys}\left[3\right]≔\left[\mathrm{diff}\left(f\left(x\,y\right)\,y\,y\right)=0\,-6\mathrm{diff}\left(f\left(x\,y\right)\,y\right)y+\mathrm{diff}\left(g\left(x\,y\right)\,y\,y\right)-2\mathrm{diff}\left(f\left(x\,y\right)\,x\,y\right)=0\,-9\mathrm{diff}\left(f\left(x\,y\right)\,y\right)a{y}^2-12\mathrm{diff}\left(f\left(x\,y\right)\,y\right)a^{2}y-3\mathrm{diff}\left(f\left(x\,y\right)\,y\right)b-3\mathrm{diff}\left(f\left(x\,y\right)\,x\right)y+2\mathrm{diff}\left(g\left(x\,y\right)\,x\,y\right)-\mathrm{diff}\left(f\left(x\,y\right)\,x\,x\right)-3\exp \left(x\right)-3g\left(x\,y\right)=0\,-6\left(\exp \left(x\right)+g\left(x\,y\right)\right)ay-6\mathrm{diff}\left(f\left(x\,y\right)\,x\right)a{y}^2-8\mathrm{diff}\left(f\left(x\,y\right)\,x\right)a^{2}y+3\mathrm{diff}\left(g\left(x\,y\right)\,y\right)a{y}^2+4\mathrm{diff}\left(g\left(x\,y\right)\,y\right)a^{2}y-4\left(\exp \left(x\right)+g\left(x\,y\right)\right)a^2-2\mathrm{diff}\left(f\left(x\,y\right)\,x\right)b+\mathrm{diff}\left(g\left(x\,y\right)\,y\right)b-3y\left(\exp \left(x\right)+\mathrm{diff}\left(g\left(x\,y\right)\,x\right)\right)+\exp \left(x\right)+\mathrm{diff}\left(g\left(x\,y\right)\,x\,x\right)=0\right]$

${\mathrm{sys}}_3≔\left[f_{y,y}=0\,-6f_{y}y+g_{y,y}-2f_{x,y}=0\,-9f_{y}a{y}^2-12f_y{a}^{2}y-3f_{y}b-3f_{x}y+2g_{x,y}-f_{x,x}-3{\ⅇ}^x-3g=0\,-6\left({\ⅇ}^x+g\right)ay-6f_{x}a{y}^2-8f_x{a}^{2}y+3g_{y}a{y}^2+4g_y{a}^{2}y-4\left({\ⅇ}^x+g\right)a^2-2f_{x}b+g_{y}b-3y\left({\ⅇ}^x+g_x\right)+{\ⅇ}^x+g_{x,x}=0\right]$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[3\right]\right)$

$\mathrm{true}$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[3\right]\,\mathrm{outputthesystem}\right)$

$\left[g=-\mathrm{\_F1}\left(x\,y\right)\,f_x=0\,f_y=0\right],\left[\right],\left[\mathrm{\_F1}\left(x\,y\right)={\ⅇ}^x\right]$

$\mathrm{ConsistencyTest}\left(\mathrm{sys}\left[3\right]\,\mathrm{outputthesystem}\,\mathrm{no\_Fn}\right)$

$\left[g=-{\ⅇ}^x\,f_x=0\,f_y=0\right]\mathbf{where}\left[\right]$

$\mathrm{dpolyform}\left(\mathrm{sys}\left[3\right]\,\mathrm{no\_Fn}\right)$

$\left[f_x=0\,f_y=0\,g_x=g\,g_y=0\right]\mathbf{where}\left[g\ne 0\right]$