algebraicsubs - Maple Help

gfun

 algebraicsubs
 substitute an algebraic function into a holonomic one

 Calling Sequence algebraicsubs(deq, eq, y(z), ini)

Parameters

 deq - linear differential equation in y(z) with polynomial coefficients eq - algebraic equation in y(z) y - name; holonomic function name z - name; variable of the holonomic function y ini - (optional) set; specify computation of initial conditions for the resulting differential equation

Description

 • The gfun[algebraicsubs](deq, eq, y(z)) command returns a differential equation satisfied by the composition $f@g$ where f is the holonomic function defined by the equation deq and g is the algebraic equation defined by eq.  The composition is holonomic by closure properties of holonomic functions.
 • Let d1 be the differential order of deq, and d2 be the degree of eq. If the equation deq is homogeneous, then the order of $f@g$ is at most $\mathrm{d1}\mathrm{d2}$.  Otherwise, it is at most $\left(\mathrm{d1}+1\right)\mathrm{d2}$.
 • If initial conditions are specified using ini, the algebraicsubs function attempts to compute initial conditions for the resulting differential equation.
 Initial conditions can be present in the differential equation if a set is specified, in the same way as initial conditions are specified for dsolve. In the case of a polynomial equation, they are specified in the optional parameter ini as a set, using the same syntax as for dsolve.

Examples

The differential equation satisfied by cos(t).

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{deq}≔{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(t\right)+f\left(t\right):$

The algebraic equation satisfied by sqrt(1-4*t).

 > $\mathrm{eq}≔\mathrm{algfuntoalgeq}\left(\sqrt{1-4t},f\left(t\right)\right):$

The differential equation satisfied by cos(sqrt(1-4*t)).

 > $\mathrm{algebraicsubs}\left(\mathrm{deq},\mathrm{eq},f\left(t\right)\right)$
 ${-}{4}{f}{}\left({t}\right){+}{2}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({t}\right){+}\left({-}{1}{+}{4}{t}\right)\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({t}\right)\right)$ (1)
 > $\mathrm{algebraicsubs}\left(\left\{{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)+y\left(x\right),y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=0\right\},2{x}^{4}-{y}^{2}+2yx-{x}^{2},y\left(x\right),\left\{y\left(0\right)=0,\mathrm{D}\left(y\right)\left(0\right)=1,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=2\sqrt{2}\right\}\right)$
 $\left\{\left({-}{2048}{{x}}^{{8}}{+}{768}{{x}}^{{6}}{+}{864}{{x}}^{{4}}{-}{236}{{x}}^{{2}}{+}{87}\right){y}{}\left({x}\right){+}\left({320}{{x}}^{{3}}{+}{280}{x}\right)\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{512}{{x}}^{{6}}{-}{136}{{x}}^{{2}}{+}{18}\right)\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({128}{{x}}^{{3}}{-}{8}{x}\right)\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{32}{{x}}^{{4}}{+}{4}{{x}}^{{2}}{+}{3}\right)\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){,}{y}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{0}{,}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({0}\right){=}{-1}{,}{{\mathrm{D}}}^{\left({3}\right)}{}\left({y}\right){}\left({0}\right){=}{-}{6}\sqrt{{2}}\right\}$ (2)