Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

The function essential_components returns a minimal characteristic decomposition of the radical  differential ideal generated by the single differential polynomial p.

Each of the characterizable components returned has a characteristic set consisting of only one differential polynomial, say $\mathrm{a1},\mathrm{...},\mathrm{ak}$.

This means that the set of solutions of the differential equation $p=0$ is minimally described as the union of the general solutions of $\mathrm{a1}=0$, ... , $\mathrm{ak}=0$.

The set of irreducible factors of $\mathrm{a1},\mathrm{...},\mathrm{ak}$ does not depend on the ranking chosen for R.

This function proceeds by eliminating the redundancy in the characteristic decomposition computed by Rosenfeld_Groebner applied to ([p], R).

The command with(diffalg,essential_components) allows the use of the abbreviated form of this command.

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

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{ranking}=\left[y\right]\,\mathrm{notation}=\mathrm{diff}\right)\:$

$p≔{\mathrm{diff}\left(y\left(t\right)\,t\right)}^3-4ty\left(t\right)\mathrm{diff}\left(y\left(t\right)\,t\right)+8{y\left(t\right)}^2$

$p≔{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+8{y\left(t\right)}^2$

$\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)\right)$

$\left[\left[{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+8{y\left(t\right)}^2\right]\,\left[27y\left(t\right)-4t^3\right]\,\left[y\left(t\right)\right]\right]$

$\mathrm{equations}\left(\mathrm{essential\_components}\left(p\,R\right)\right)$

$\left[\left[{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+8{y\left(t\right)}^2\right]\,\left[27y\left(t\right)-4t^3\right]\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{ranking}=\left[y\right]\right)\:$

$p≔{y\left[t\right]}^2-4y\left[\right]$

$p≔y_t^2-4y\left[\right]$

$q≔{y\left[t\right]}^2-4{y\left[\right]}^3$

$q≔-4{y\left[\right]}^3+y_t^2$

$\mathrm{Cp}≔\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)\right)$

$\mathrm{Cp}≔\left[\left[y_t^2-4y\left[\right]\right]\,\left[y\left[\right]\right]\right]$

$\mathrm{Cq}≔\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[q\right]\,R\right)\right)$

$\mathrm{Cq}≔\left[\left[-4{y\left[\right]}^3+y_t^2\right]\,\left[y\left[\right]\right]\right]$

$\mathrm{Mp}≔\mathrm{equations}\left(\mathrm{essential\_components}\left(p\,R\right)\right)$

$\mathrm{Mp}≔\left[\left[y_t^2-4y\left[\right]\right]\,\left[y\left[\right]\right]\right]$

$\mathrm{Mq}≔\mathrm{equations}\left(\mathrm{essential\_components}\left(q\,R\right)\right)$

$\mathrm{Mq}≔\left[\left[-4{y\left[\right]}^3+y_t^2\right]\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[u\right]\right)\:$

$p≔-u\left[\right]+yu\left[y\right]+xu\left[x\right]-{u\left[x\right]}^2-{u\left[y\right]}^2$

$p≔x{u}_x+y{u}_y-u_x^2-u_y^2-u\left[\right]$

$C≔\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)\right)$

$C≔\left[\left[-x{u}_x-y{u}_y+u_x^2+u_y^2+u\left[\right]\right]\,\left[2u_x-x\,-x^2-4y{u}_y+4u_y^2+4u\left[\right]\right]\,\left[-x^2-y^2+4u\left[\right]\right]\right]$

$M≔\mathrm{equations}\left(\mathrm{essential\_components}\left(p\,R\right)\right)$

$M≔\left[\left[-x{u}_x-y{u}_y+u_x^2+u_y^2+u\left[\right]\right]\,\left[-x^2-y^2+4u\left[\right]\right]\right]$

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

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[u\,v\right]\right)\:$

$p≔{u\left[x,y\right]}^{2}v\left[y\right]-u\left[x,y\right]v\left[y\right]u\left[y\right]-u\left[y\right]u\left[x,y\right]+{u\left[y\right]}^2$

$p≔-u_{x,y}{v}_y{u}_y+u_{x,y}^2{v}_y+u_y^2-u_y{u}_{x,y}$

$\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)$

$\left[\mathrm{characterizable}\,\mathrm{characterizable}\right]$

$\mathrm{MR}≔\mathrm{essential\_components}\left(p\,R\right)$

$\mathrm{MR}≔\left[\mathrm{characterizable}\right]$

$\mathrm{ER}≔\mathrm{equations}\left(\mathrm{MR}\right)$

$\mathrm{ER}≔\left[\left[-u_{x,y}{v}_y{u}_y+u_{x,y}^2{v}_y+u_y^2-u_y{u}_{x,y}\right]\right]$

$Q≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[v\,u\right]\right)\:$

$\mathrm{MQ}≔\mathrm{essential\_components}\left(p\,Q\right)$

$\mathrm{MQ}≔\left[\mathrm{characterizable}\right]$

$\mathrm{EQ}≔\mathrm{equations}\left(\mathrm{MQ}\right)$

$\mathrm{EQ}≔\left[\left[u_{x,y}{v}_y-u_y\right]\right]$

$\mathrm{factor}\left(\mathrm{ER}\left[1\right]\left[1\right]\right)$

$\left(-u_{x,y}+u_y\right)\left(-u_{x,y}{v}_y+u_y\right)$

$R≔\mathrm{differential\_ring}\left(\mathrm{ranking}=\left[y\right]\,\mathrm{derivations}=\left[x\right]\right)\:$

$\mathrm{chazy}≔-{\left(y\left[x,x\right]+{y\left[\right]}^{3}y\left[x\right]\right)}^2+{\left(y\left[\right]y\left[x\right]\right)}^2\left(4y\left[x\right]+{y\left[\right]}^4\right)$

$\mathrm{chazy}≔-{\left({y\left[\right]}^3{y}_x+y_{x,x}\right)}^2+{y\left[\right]}^2{y}_x^2\left({y\left[\right]}^4+4y_x\right)$

$\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{chazy}\right]\,R\right)\right)$

$\left[\left[2{y\left[\right]}^3{y}_x{y}_{x,x}-4{y\left[\right]}^2{y}_x^3+y_{x,x}^2\right]\,\left[{y\left[\right]}^4+4y_x\right]\,\left[y_x\right]\right]$

$\mathrm{equations}\left(\mathrm{essential\_components}\left(\mathrm{chazy}\,R\right)\right)$

$\left[\left[2{y\left[\right]}^3{y}_x{y}_{x,x}-4{y\left[\right]}^2{y}_x^3+y_{x,x}^2\right]\,\left[{y\left[\right]}^4+4y_x\right]\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{ranking}=\left[y\,z\right]\,\mathrm{derivations}=\left[x\right]\right)\:$

$\mathrm{rubel}≔3{y\left[x\right]}^{4}y\left[\mathrm{`\$`}\left(x\,2\right)\right]{y\left[\mathrm{`\$`}\left(x\,4\right)\right]}^2-4{y\left[x\right]}^4{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^{2}y\left[\mathrm{`\$`}\left(x\,4\right)\right]+6{y\left[x\right]}^3{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^{2}y\left[\mathrm{`\$`}\left(x\,3\right)\right]y\left[\mathrm{`\$`}\left(x\,4\right)\right]+24{y\left[x\right]}^2{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^{4}y\left[\mathrm{`\$`}\left(x\,4\right)\right]-12{y\left[x\right]}^{3}y\left[\mathrm{`\$`}\left(x\,2\right)\right]{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^3-29{y\left[x\right]}^2{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^3{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^2+12{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^7$

$\mathrm{rubel}≔3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7$

$\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{rubel}\right]\,R\right)\right)$

$\left[\left[3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7\right]\,\left[y_x^2{y}_{x,x,x}^2+3y_{x,x}^4\right]\,\left[y_{x,x}\right]\right]$

$\mathrm{equations}\left(\mathrm{essential\_components}\left(\mathrm{rubel}\,R\right)\right)$

$\left[\left[3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7\right]\,\left[y_x^2{y}_{x,x,x}^2+3y_{x,x}^4\right]\right]$