gfun[algeqtodiffeq] - compute a differential equation satisfied by an algebraic function
Calling Sequence
algeqtodiffeq(p, y(z), ini, options)
Parameters
p - polynomial in y and z (or a polynomial equation)
y - name of the holonomic function
z - name of the generic variable associated with y
ini - (optional) initial conditions to specify a solution of eq
options - (optional) equation(s) of the form homogeneous=true, ini_cond=false, and/or operator=name
Description
- The polynomial
pdefines an algebraic functiony(z). This procedure computes a linear differential equation with polynomial coefficients that it verifies. This equation is of order at mostdegree(p,y)-1. - The output contains initial conditions at zero (
y(0),D(y)(0), and so on), and can thus be given directly to dsolve. In general,y(0)is aRootOfof a polynomial,D(y)(0)a rational expression iny(0),(D@@2)(y)(0)a rational expression iny(0),D(y)(0), and so on. - If the optional argument "homoegenous=true" is given, the differential equation will be forced to be homogeneous.
- If the optional argument "ini_cond=false" is given, no attempt at computing initial conditions at 0 will be made and the equation will be returned without initial conditions.
- If
operator=nameis given, the supplied name is used verbatim as the differential operator and initial conditions are not computed. A homogeneous relation is returned as an operator polynomial; an inhomogeneous relationq+L(y)=0is returned asL(name)=-q.
Examples
> with(gfun):
algeqtodiffeq(y=1+z*y^2,y(z));\[1+\left(-1+2 z \right) y \! \left(z \right)+\left(4 z^{2}-z \right) \left(\frac{d}{d z}y \! \left(z \right)\right)\]
> algeqtodiffeq(56*a^3+7*a^3*y^3-14*y*z,y(z),{y(0)=-2});\[\left\{y \! \left(z \right) z -3 z^{2} \left(\frac{d}{d z}y \! \left(z \right)\right)+\left(108 a^{9}-2 z^{3}\right) \left(\frac{d^{2}}{d z^{2}}y \! \left(z \right)\right), y \! \left(0\right) = -2, D\! \left(y \right)\! \left(0\right) = -\frac{1}{3 a^{3}}\right\}\]
We can use algeqtodiffeq with diffeqtorec to determine fast Taylor expansions.
> p:=y=1+z*y+z*y^5;\[y = z \,y^{5}+z y +1\]
> deq:=algeqtodiffeq(p,y(z)):
diffeqtoseries(deq,y(z),10);\[1+2 z +12 z^{2}+112 z^{3}+1232 z^{4}+14832 z^{5}+189184 z^{6}+2512064 z^{7}+34358784 z^{8}+480745984 z^{9}+6848734464 z^{10}+\mathrm{O}\! \left(z^{11}\right)\]
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