gfun[holexprtodiffeq] - produce a differential equation satisfied by a holonomic expression
Calling Sequence
holexprtodiffeq(expr, y(z), <iniconds>, <homogeneous=bool>, <operator=name>)
Parameters
expr - holonomic expression in y(z)
y,z - name of the holonomic function and the generic variable
homogeneous - (optional) boolean requesting a homogeneous output equation; the default is false
operator - (optional) name used verbatim as the differential operator
Description
- If
expris a holonomic expression, thengfun[holexprtodiffeq]returns a differential equation inyandzthat is satisfied by expr. Initial conditions are given whenever possible. - If
homogeneous=true, the closure computation retains the constant coordinate and returns a homogeneous differential equation. In particular, a constant expression is represented by a first-order homogeneous equation. - With
operator=name, initial conditions are not computed. Homogeneous output is a polynomial in the supplied name and inhomogeneous output has the formL(name)=-q. - Not all holonomic functions are recognized by
holexprtodiffeq. It currently knows about the following functions:
BesselI BesselJ BesselK BesselY arccos arccosh arccot arccoth arccsc arccsch arcsec arcsech arcsin arcsinh arctan arctanh cos cosh erf erfc exp ln sin sinh
Examples
> with(gfun):
holexprtodiffeq(BesselJ(2,x),y(x));\[\left\{x^{2} \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+y \! \left(x \right) x^{2}+x \left(\frac{d}{d x}y \! \left(x \right)\right)-4 y \! \left(x \right), D^{\left(2\right)}\! \left(y \right)\! \left(0\right) = {\frac{1}{4}}\right\}\]
> holexprtodiffeq(arcsec(1/x)+sin(x)^2,y(x));\[\left\{\left(4 x^{6}-10 x^{4}+11 x^{2}-5\right) \left(\frac{d^{4}}{d x^{4}}y \! \left(x \right)\right)+\left(4 x^{5}-2 x^{3}+13 x \right) \left(\frac{d^{3}}{d x^{3}}y \! \left(x \right)\right)+\left(16 x^{6}-40 x^{4}+44 x^{2}-20\right) \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+\left(16 x^{5}-8 x^{3}+52 x \right) \left(\frac{d}{d x}y \! \left(x \right)\right), y \! \left(0\right) = \frac{\pi}{2}, D\! \left(y \right)\! \left(0\right) = -1, D^{\left(2\right)}\! \left(y \right)\! \left(0\right) = 2, D^{\left(3\right)}\! \left(y \right)\! \left(0\right) = -1\right\}\]