hypergeomsols - Maple Help

DEtools

 hypergeomsols
 solutions of a hypergeometric type second order linear ODE

 Calling Sequence hypergeomsols(LODE) hypergeomsols(LODE,y(x)) hypergeomsols(coeff_list,x)

Parameters

 LODE - homogeneous linear differential equation of second order y(x) - any indeterminate function of one variable; required only when the ODE involves more than one function being differentiated coeff_list - list of coefficients of the linear ODE x - independent variable of the linear ODE

Description

 • The hypergeomsols routine returns a basis of the space of solutions of a second order linear ODE of hypergeometric type--that is, an equation of the form (see hyperode)
 > DEtools[hyperode](hypergeom([a,b], [c], x), y(x)) = 0;
 ${y}{}\left({x}\right){}{a}{}{b}{+}\left(\left({a}{+}{b}{+}{1}\right){}{x}{-}{c}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (1)
 For making symbolic experiments with the hypergeom function and the differential equation it satisfies see also dpolyform.
 • There are two general forms of calling sequences available for hypergeomsols.
 The first argument LODE of the first calling sequence is a linear differential equation in diff or $\mathrm{D}$ form. The second argument y(x) is the function in the differential equation, and it is required only when the ODE involves more than one function being differentiated.
 The last calling sequence has as its first argument the list of coefficients of a linear ODE, and the second argument is the independent variable. This input sequence may be convenient for programming with the hypergeomsols routine.
 • This routine is part of the DEtools package, and so it can be used in the form hypergeomsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[hypergeomsols](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(y\left(x\right),\mathrm{prime}=x\right)$
 ${y}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{y}$
 ${\mathrm{derivatives with respect to}}{}{x}{}{\mathrm{of functions of one variable will now be displayed with \text{'}}}$ (2)

Kamke's example 2.108:

 > $\mathrm{ode}\left[108\right]≔x\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(x+a+b\right)\mathrm{diff}\left(y\left(x\right),x\right)+ay\left(x\right)$
 ${{\mathrm{ode}}}_{{108}}{≔}{x}{}{\mathrm{y\text{'}\text{'}}}{+}\left({x}{+}{a}{+}{b}\right){}{\mathrm{y\text{'}}}{+}{a}{}{y}$ (3)
 > $B\left[108\right]≔\mathrm{hypergeomsols}\left(\mathrm{ode}\left[108\right],y\left(x\right)\right)$
 ${{B}}_{{108}}{≔}\left[{\mathrm{KummerM}}{}\left({a}{,}{a}{+}{b}{,}{-}{x}\right){,}{\mathrm{KummerU}}{}\left({a}{,}{a}{+}{b}{,}{-}{x}\right)\right]$ (4)

A solution to this ODE is built by using this solution basis and tested as follows.

 > $\mathrm{sol}\left[108\right]≔y\left(x\right)=\mathrm{add}\left(\mathrm{_C}‖iB\left[108\right]\left[i\right],i=1..2\right)$
 ${{\mathrm{sol}}}_{{108}}{≔}{y}{=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left({a}{,}{a}{+}{b}{,}{-}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left({a}{,}{a}{+}{b}{,}{-}{x}\right)$ (5)
 > $\mathrm{odetest}\left(\mathrm{sol}\left[108\right],\mathrm{ode}\left[108\right]\right)$
 ${0}$ (6)

Kamke's example 2.114:

 > $\mathrm{ode}\left[114\right]≔-y\left(x\right)+\mathrm{diff}\left(y\left(x\right),x\right)\left(-2x+2\right)+x\mathrm{diff}\left(y\left(x\right),x,x\right)$
 ${{\mathrm{ode}}}_{{114}}{≔}{-}{y}{+}{\mathrm{y\text{'}}}{}\left({-}{2}{}{x}{+}{2}\right){+}{x}{}{\mathrm{y\text{'}\text{'}}}$ (7)
 > $B\left[114\right]≔\mathrm{hypergeomsols}\left(\mathrm{ode}\left[114\right],y\left(x\right)\right)$
 ${{B}}_{{114}}{≔}\left[{{ⅇ}}^{{2}{}{x}}{}{\mathrm{KummerM}}{}\left(\frac{{3}}{{2}}{,}{2}{,}{-}{2}{}{x}\right){,}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{KummerU}}{}\left(\frac{{3}}{{2}}{,}{2}{,}{-}{2}{}{x}\right)\right]$ (8)

Kamke's example 2.116:

 > $\mathrm{ode}\left[116\right]≔x\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\left(ax+b+n\right)\mathrm{diff}\left(y\left(x\right),x\right)+nay\left(x\right)$
 ${{\mathrm{ode}}}_{{116}}{≔}{x}{}{\mathrm{y\text{'}\text{'}}}{+}\left({a}{}{x}{+}{b}{+}{n}\right){}{\mathrm{y\text{'}}}{+}{n}{}{a}{}{y}$ (9)
 > $B\left[116\right]≔\mathrm{hypergeomsols}\left(\mathrm{ode}\left[116\right],y\left(x\right)\right)$
 ${{B}}_{{116}}{≔}\left[{\mathrm{KummerM}}{}\left({n}{,}{b}{+}{n}{,}{-}{a}{}{x}\right){,}{\mathrm{KummerU}}{}\left({n}{,}{b}{+}{n}{,}{-}{a}{}{x}\right)\right]$ (10)