holexprtodiffeq - Maple Help

gfun

 holexprtodiffeq
 produce a differential equation satisfied by a holonomic expression

 Calling Sequence holexprtodiffeq(expr, y(z))

Parameters

 expr - holonomic expression in y(z) y - name; holonomic function name z - name; variable of the holonomic function y

Description

 • The gfun[holexprtodiffeq](expr, y(z)) command produces a differential equation satisfied by the holonomic expression expr.
 If expr is a holonomic expression, then the gfun[holexprtodiffeq] function returns a differential equation in y and z that is satisfied by expr. Initial conditions are returned when possible.
 • Not all holonomic functions are recognized by holexprtodiffeq. It currently accepts 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

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{holexprtodiffeq}\left(\mathrm{BesselJ}\left(2,x\right),y\left(x\right)\right)$
 $\left\{{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{4}\right){}{y}{}\left({x}\right){,}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({0}\right){=}\frac{{1}}{{4}}\right\}$ (1)
 > $\mathrm{holexprtodiffeq}\left(\mathrm{arcsec}\left(\frac{1}{x}\right)+{\mathrm{sin}\left(x\right)}^{2},y\left(x\right)\right)$
 $\left\{\left({16}{}{{x}}^{{5}}{-}{8}{}{{x}}^{{3}}{+}{52}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({16}{}{{x}}^{{6}}{-}{40}{}{{x}}^{{4}}{+}{44}{}{{x}}^{{2}}{-}{20}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({4}{}{{x}}^{{5}}{-}{2}{}{{x}}^{{3}}{+}{13}{}{x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({4}{}{{x}}^{{6}}{-}{10}{}{{x}}^{{4}}{+}{11}{}{{x}}^{{2}}{-}{5}\right){}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){,}{y}{}\left({0}\right){=}\frac{{\mathrm{\pi }}}{{2}}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{-1}{,}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({0}\right){=}{2}{,}{{\mathrm{D}}}^{\left({3}\right)}{}\left({y}\right){}\left({0}\right){=}{-1}\right\}$ (2)