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

The initial conditions of J are the derivatives that are not derivatives of any leader of its equations (J).

If order is present, the function initial_conditions returns the list of the initial conditions of J of order less than order.

If order is omitted and there are only finitely many initial conditions, the function returns them all.

If order is omitted and there are infinitely many initial conditions, FAIL is returned.

The initial conditions appear in the terms of the power series computed for J.

If J is a radical differential ideal presented by a list of characterizable differential ideals  then the function is mapped on all its components.

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

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

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

$\mathrm{p1}≔v\left[\right]{u\left[x,x\right]}^2+v\left[\right]u\left[x,x\right]+u\left[x\right]\:$

$\mathrm{p2}≔u\left[x,y\right]\:$

$\mathrm{p3}≔{u\left[y,y\right]}^2-1\:$

$J≔\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\,\left[u\left[x\right]\right]\,R\right)\:$

$\mathrm{rewrite\_rules}\left(J\right)$

$\left[\left[u_{x,x}^2=-\frac{v\left[\right]u_{x,x}+u_x}{v\left[\right]}\,u_{x,y}=0\,u_{y,y}^2=1\,v_y=0\right]\,\left[u_{y,y}^2=1\,u_x=\frac{v\left[\right]}{4}\,v_x=\mathrm{-2}\,v_y=0\right]\right]$

$\mathrm{initial\_conditions}\left(J\right)$

$\left[\mathrm{FAIL}\,\left[\mathrm{\_Cu}\,\mathrm{\_Cv}\,\mathrm{\_Cu\_y}\right]\right]$