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.

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

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

$R≔\mathrm{ODE\_ring}$

$p≔\left(u\left[t\right]-1\right){u\left[t,t\right]}^2+u\left[t,t\right]+{u\left[t\right]}^2+tu\left[\right]$

$p≔\left(u_t-1\right)u_{t,t}^2+u_{t,t}+u_t^2+tu\left[\right]$

$\mathrm{leader}\left(p\,R\right)$

$u_{t,t}$

$\mathrm{rank}\left(p\,R\right)$

$u_{t,t}^2$

$\mathrm{initial}\left(p\,R\right)$

$u_t-1$

$\mathrm{separant}\left(p\,R\right)$

$2u_t{u}_{t,t}-2u_{t,t}+1$

$K≔\mathrm{field\_extension}\left(\mathrm{transcendental\_elements}=\left[a\,b\right]\right)\:$

$R≔\mathrm{differential\_ring}\left(\mathrm{field\_of\_constants}=K\,\mathrm{derivations}=\left[t\right]\,\mathrm{ranking}=\left[u\right]\right)$

$R≔\mathrm{ODE\_ring}$

$p≔2u\left[t\right]b{u\left[t,t\right]}^2+\frac{u\left[\right]{u\left[t,t\right]}^2}{a}+u\left[t\right]$

$p≔2u_{t}b{u}_{t,t}^2+\frac{u\left[\right]u_{t,t}^2}{a}+u_t$

$\mathrm{initial}\left(p\,R\right)$

$\frac{2ab{u}_t+u\left[\right]}{a}$

$\mathrm{leader}\left(p\,R\right)$

$u_{t,t}$

$\mathrm{rank}\left(p\,R\right)$

$u_{t,t}^2$

$\mathrm{initial}\left(p\,R\right)$

$\frac{2ab{u}_t+u\left[\right]}{a}$

$\mathrm{separant}\left(p\,R\right)$

$\frac{4ab{u}_t{u}_{t,t}+2u\left[\right]u_{t,t}}{a}$

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

$p≔u_{x,y}{v}_{x,y,y}^2+u_{x,x}^2+u_x$

$\mathrm{R1}≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[u\,v\right]\right)$

$\mathrm{R1}≔\mathrm{PDE\_ring}$

$\mathrm{leader}\left(p\,\mathrm{R1}\right)$

$u_{x,x}$

$\mathrm{rank}\left(p\,\mathrm{R1}\right)$

$u_{x,x}^2$

$\mathrm{initial}\left(p\,\mathrm{R1}\right)$

$1$

$\mathrm{separant}\left(p\,\mathrm{R1}\right)$

$2u_{x,x}$

$\mathrm{initial}\left(\mathrm{differentiate}\left(p\,x\,x\,\mathrm{R1}\right)\,\mathrm{R1}\right)$

$2u_{x,x}$

$\mathrm{R2}≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\left[u\,v\right]\right]\right)$

$\mathrm{R2}≔\mathrm{PDE\_ring}$

$\mathrm{leader}\left(p\,\mathrm{R2}\right)$

$v_{x,y,y}$

$\mathrm{rank}\left(p\,\mathrm{R2}\right)$

$v_{x,y,y}^2$

$\mathrm{initial}\left(p\,\mathrm{R2}\right)$

$u_{x,y}$

$\mathrm{separant}\left(p\,\mathrm{R2}\right)$

$2v_{x,y,y}{u}_{x,y}$

$\mathrm{initial}\left(\mathrm{differentiate}\left(p\,x\,x\,\mathrm{R2}\right)\,\mathrm{R2}\right)$

$2v_{x,y,y}{u}_{x,y}$