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 symmetric_power
 calculate the symmetric power of a differential equation or operator

 Calling Sequence symmetric_power(L, m, domain) symmetric_power(eqn, m, dvar)

Parameters

 L - differential operator m - positive integer domain - list containing two names eqn - homogeneous linear differential equation dvar - dependent variable

Description

 • The input L is a differential operator. The output of this procedure is a linear differential operator M of minimal order such that for every set of m solutions $\mathrm{y1},\mathrm{...},\mathrm{ym}$ of L the product $\mathrm{y1}\mathrm{y2}\dots \mathrm{ym}$ is a solution of $M$.
 • The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt},t\right]$, then the differential operators are notated with the symbols $\mathrm{Dt}$ and $t$. They are viewed as elements of the differential algebra $C\left(t\right)$ $\left[\mathrm{Dt}\right]$ where $C$ is the field of constants.
 • If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.
 • Instead of a differential operator, the input can also be a linear homogeneous differential equation having rational function coefficients. In this case the third argument must be the dependent variable.
 • This function is part of the DEtools package, and so it can be used in the form symmetric_power(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[symmetric_power](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{_Envdiffopdomain}≔\left[\mathrm{Dx},x\right]:$
 > $L≔{\mathrm{Dx}}^{2}+a\left(x\right)\mathrm{Dx}+b\left(x\right)$
 ${L}{≔}{{\mathrm{Dx}}}^{{2}}{+}{a}{}\left({x}\right){}{\mathrm{Dx}}{+}{b}{}\left({x}\right)$ (1)
 > $M≔\mathrm{symmetric_power}\left(L,2\right)$
 ${M}{≔}{{\mathrm{Dx}}}^{{3}}{+}{3}{}{a}{}\left({x}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({2}{}{{a}{}\left({x}\right)}^{{2}}{+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{a}{}\left({x}\right){+}{4}{}{b}{}\left({x}\right)\right){}{\mathrm{Dx}}{+}{4}{}{b}{}\left({x}\right){}{a}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{b}{}\left({x}\right)$ (2)

To illustrate formally the meaning of the output of this command, consider a general second order ODE

 > $\mathrm{ODE}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+a\left(x\right)y\left(x\right)=0$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{a}{}\left({x}\right){}{y}{}\left({x}\right){=}{0}$ (3)

The nth symmetric_power of ODE is another ODE having for a solution the nth power of the solution of ODE. For example, the solution of ODE can be written - formally - using the Maple DESol command; dsolve represents it that way:

 > $\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$
 ${\mathrm{sol}}{≔}{y}{}\left({x}\right){=}{\mathrm{DESol}}{}\left(\left\{{a}{}\left({x}\right){}{\mathrm{_Y}}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({x}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({x}\right)\right\}\right)$ (4)

where in the above DESol(...) represents any linear combination of two independent solutions of ODE. The first symmetric power of ODE is then ODE itself (has for solution sol^1) and, for instance, for the second and third symmetric powers of ODE we have

 > $\mathrm{ODE_2}≔\mathrm{symmetric_power}\left(\mathrm{ODE},2,y\left(x\right)\right)$
 ${\mathrm{ODE_2}}{≔}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{a}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{4}{}{a}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ODE_2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{{\mathrm{DESol}}{}\left(\left\{{a}{}\left({x}\right){}{\mathrm{_Y}}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({x}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({x}\right)\right\}\right)}^{{2}}$ (6)
 > $\mathrm{ODE_3}≔\mathrm{symmetric_power}\left(\mathrm{ODE},3,y\left(x\right)\right)$
 ${\mathrm{ODE_3}}{≔}\left({9}{}{{a}{}\left({x}\right)}^{{2}}{+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{a}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{10}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{a}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{10}{}{a}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (7)
 > $\mathrm{dsolve}\left(\mathrm{ODE_3},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{{\mathrm{DESol}}{}\left(\left\{{a}{}\left({x}\right){}{\mathrm{_Y}}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({x}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({x}\right)\right\}\right)}^{{3}}$ (8)