diffalg

 return the leader of a differential polynomial
 rank
 return the rank of a differential polynomial
 initial
 return the initial of a differential polynomial
 separant
 return the separant of a differential polynomial

 Calling Sequence leader (q, R) rank (q, R) initial (q, R) separant (q, R)

Parameters

 q - differential polynomial R - differential polynomial ring

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The leader of a differential polynomial q is the greatest derivative occurring in q with respect to the ranking of R.
 • The rank of q is the leader of q raised to the degree of q with respect to its leader.
 • The initial of q is the leading coefficient of q with respect to its leader.
 • The separant of q is the partial derivative of q with respect to its leader. It is also the initial of any proper derivative of q.
 • If q belongs to the ground field R, then leader, rank, initial and separant return an error message.
 • The command with(diffalg,leader) allows the use of the abbreviated form of this command.
 • The command with(diffalg,rank) allows the use of the abbreviated form of this command.
 • The command with(diffalg,initial) allows the use of the abbreviated form of this command.
 • The command with(diffalg,separant) allows the use of the abbreviated form of this command.

Examples

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

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[t\right],\mathrm{ranking}=\left[u\right]\right)$
 ${R}{≔}{\mathrm{ODE_ring}}$ (1)
 > $p≔\left(u\left[t\right]-1\right){u\left[t,t\right]}^{2}+u\left[t,t\right]+{u\left[t\right]}^{2}+tu\left[\right]$
 ${p}{≔}\left({{u}}_{{t}}{-}{1}\right){}{{u}}_{{t}{,}{t}}^{{2}}{+}{{u}}_{{t}{,}{t}}{+}{{u}}_{{t}}^{{2}}{+}{t}{}{u}\left[\right]$ (2)
 > $\mathrm{leader}\left(p,R\right)$
 ${{u}}_{{t}{,}{t}}$ (3)
 > $\mathrm{rank}\left(p,R\right)$
 ${{u}}_{{t}{,}{t}}^{{2}}$ (4)
 > $\mathrm{initial}\left(p,R\right)$
 ${{u}}_{{t}}{-}{1}$ (5)
 > $\mathrm{separant}\left(p,R\right)$
 ${2}{}{{u}}_{{t}}{}{{u}}_{{t}{,}{t}}{-}{2}{}{{u}}_{{t}{,}{t}}{+}{1}$ (6)

 > $K≔\mathrm{field_extension}\left(\mathrm{transcendental_elements}=\left[a,b\right]\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{field_of_constants}=K,\mathrm{derivations}=\left[t\right],\mathrm{ranking}=\left[u\right]\right)$
 ${R}{≔}{\mathrm{ODE_ring}}$ (7)
 > $p≔2u\left[t\right]b{u\left[t,t\right]}^{2}+\frac{u\left[\right]{u\left[t,t\right]}^{2}}{a}+u\left[t\right]$
 ${p}{≔}{2}{}{{u}}_{{t}}{}{b}{}{{u}}_{{t}{,}{t}}^{{2}}{+}\frac{{u}\left[\right]{}{{u}}_{{t}{,}{t}}^{{2}}}{{a}}{+}{{u}}_{{t}}$ (8)
 > $\mathrm{initial}\left(p,R\right)$
 $\frac{{2}{}{a}{}{b}{}{{u}}_{{t}}{+}{u}\left[\right]}{{a}}$ (9)
 > $\mathrm{leader}\left(p,R\right)$
 ${{u}}_{{t}{,}{t}}$ (10)
 > $\mathrm{rank}\left(p,R\right)$
 ${{u}}_{{t}{,}{t}}^{{2}}$ (11)
 > $\mathrm{initial}\left(p,R\right)$
 $\frac{{2}{}{a}{}{b}{}{{u}}_{{t}}{+}{u}\left[\right]}{{a}}$ (12)
 > $\mathrm{separant}\left(p,R\right)$
 $\frac{{4}{}{a}{}{b}{}{{u}}_{{t}}{}{{u}}_{{t}{,}{t}}{+}{2}{}{u}\left[\right]{}{{u}}_{{t}{,}{t}}}{{a}}$ (13)

 > $p≔u\left[x,y\right]{v\left[x,y,y\right]}^{2}+{u\left[x,x\right]}^{2}+u\left[x\right]$
 ${p}{≔}{{u}}_{{x}{,}{y}}{}{{v}}_{{x}{,}{y}{,}{y}}^{{2}}{+}{{u}}_{{x}{,}{x}}^{{2}}{+}{{u}}_{{x}}$ (14)
 > $\mathrm{R1}≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u,v\right]\right)$
 ${\mathrm{R1}}{≔}{\mathrm{PDE_ring}}$ (15)
 > $\mathrm{leader}\left(p,\mathrm{R1}\right)$
 ${{u}}_{{x}{,}{x}}$ (16)
 > $\mathrm{rank}\left(p,\mathrm{R1}\right)$
 ${{u}}_{{x}{,}{x}}^{{2}}$ (17)
 > $\mathrm{initial}\left(p,\mathrm{R1}\right)$
 ${1}$ (18)
 > $\mathrm{separant}\left(p,\mathrm{R1}\right)$
 ${2}{}{{u}}_{{x}{,}{x}}$ (19)
 > $\mathrm{initial}\left(\mathrm{differentiate}\left(p,x,x,\mathrm{R1}\right),\mathrm{R1}\right)$
 ${2}{}{{u}}_{{x}{,}{x}}$ (20)
 > $\mathrm{R2}≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[\left[u,v\right]\right]\right)$
 ${\mathrm{R2}}{≔}{\mathrm{PDE_ring}}$ (21)
 > $\mathrm{leader}\left(p,\mathrm{R2}\right)$
 ${{v}}_{{x}{,}{y}{,}{y}}$ (22)
 > $\mathrm{rank}\left(p,\mathrm{R2}\right)$
 ${{v}}_{{x}{,}{y}{,}{y}}^{{2}}$ (23)
 > $\mathrm{initial}\left(p,\mathrm{R2}\right)$
 ${{u}}_{{x}{,}{y}}$ (24)
 > $\mathrm{separant}\left(p,\mathrm{R2}\right)$
 ${2}{}{{v}}_{{x}{,}{y}{,}{y}}{}{{u}}_{{x}{,}{y}}$ (25)
 > $\mathrm{initial}\left(\mathrm{differentiate}\left(p,x,x,\mathrm{R2}\right),\mathrm{R2}\right)$
 ${2}{}{{v}}_{{x}{,}{y}{,}{y}}{}{{u}}_{{x}{,}{y}}$ (26)