FindODE - Maple Help

DEtools

 FindODE
 find a linear ODE for a holonomic function

 Calling Sequence FindODE(f, v, maxorder)

Parameters

 f - expression v - either the dependent variable, of the form $y\left(x\right)$, or a list of two names $\left[\mathrm{Dx},x\right]$ maxorder - (optional) posint; maximal order of the ODE. Default value: $6$

Description

 • The input f should be a holonomic function, in other words, a function that satisfies a linear ODE with rational function coefficients. The FindODE function tries to find such an ODE.
 • If FindODE fails to find a linear ODE, then $\mathrm{FAIL}$ is returned.
 • When using the $\mathrm{FindODE}\left(f,y\left(x\right)\right)$ calling sequence, the ODE will be returned in terms of the dependent variable $y\left(x\right)$, where $x$ is the independent variable.
 • When using the $\mathrm{FindODE}\left(f,\left[\mathrm{Dx},x\right]\right)$ calling sequence, the result will be given in differential operator form, that is, as a polynomial in the differential operator $\mathrm{Dx}$ whose coefficients are polynomials in the independent variable $x$.
 • The second argument $v$ can be omitted if the environment variable $\mathrm{_Envdiffopdomain}$ is assigned a list of two names, in which case the result will be given in differential operator notation.
 • The resulting ODE will be cleared of denominators, that is, its coefficients are polynomials in the independent variable $x$ without common factors.
 • By default, FindODE incrementally searches for an ODE up to order 6. This maximal order can be overridden by specifying the optional third argument, maxorder.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{FindODE}\left(\mathrm{cos}\left(\mathrm{sqrt}\left(x\right)\right),y\left(x\right)\right)$
 ${y}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1)
 > $\mathrm{FindODE}\left(\mathrm{exp}\left(x\right)+\mathrm{sqrt}\left(x\right),y\left(x\right)\right)$
 $\left({2}{}{x}{+}{1}\right){}{y}{}\left({x}\right){+}\left({-}{4}{}{{x}}^{{2}}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({4}{}{{x}}^{{2}}{-}{2}{}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (2)

The hypergeometric 2F1 equation.

 > $\mathrm{FindODE}\left(\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],x\right),y\left(x\right)\right)$
 ${a}{}{b}{}{y}{}\left({x}\right){+}\left({x}{}{a}{+}{x}{}{b}{-}{c}{+}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (3)

The tangent function is not holonomic, so the result is $\mathrm{FAIL}$.

 > $\mathrm{FindODE}\left(\mathrm{tan}\left(x\right),y\left(x\right)\right)$
 ${\mathrm{FAIL}}$ (4)

Operator notation.

 > $\mathrm{FindODE}\left(\mathrm{cos}\left(\mathrm{sqrt}\left(x\right)\right),\left[\mathrm{Dx},x\right]\right)$
 ${4}{}{x}{}{{\mathrm{Dx}}}^{{2}}{+}{2}{}{\mathrm{Dx}}{+}{1}$ (5)

The following example is a generating function from the Online Encyclopedia of Integer Sequences (http://oeis.org/A151357).

 > $\mathrm{ogf}≔{x}^{-2}\left(\mathrm{Int}\left(\mathrm{Int}\left(\frac{2\mathrm{hypergeom}\left(\left[\frac{3}{4},\frac{5}{4}\right],\left[2\right],\frac{64{x}^{3}\left(1+x\right)}{{\left(1-4{x}^{2}\right)}^{2}}\right)}{{\left(1-4{x}^{2}\right)}^{\frac{3}{2}}},x\right),x\right)\right)$
 ${\mathrm{ogf}}{≔}\frac{{\int }{\int }\frac{{2}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{3}}{{4}}{,}\frac{{5}}{{4}}\right]{,}\left[{2}\right]{,}\frac{{64}{}{{x}}^{{3}}{}\left({x}{+}{1}\right)}{{\left({-}{4}{}{{x}}^{{2}}{+}{1}\right)}^{{2}}}\right)}{{\left({-}{4}{}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{x}}^{{2}}}$ (6)
 > $L≔\mathrm{FindODE}\left(\mathrm{ogf},\left[\mathrm{Dx},x\right]\right)$
 ${L}{≔}\left({192}{}{{x}}^{{10}}{+}{640}{}{{x}}^{{9}}{+}{880}{}{{x}}^{{8}}{+}{656}{}{{x}}^{{7}}{+}{244}{}{{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{-}{7}{}{{x}}^{{4}}{-}{3}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{4}}{+}\left({3072}{}{{x}}^{{9}}{+}{9792}{}{{x}}^{{8}}{+}{13344}{}{{x}}^{{7}}{+}{10016}{}{{x}}^{{6}}{+}{3632}{}{{x}}^{{5}}{+}{244}{}{{x}}^{{4}}{-}{86}{}{{x}}^{{3}}{-}{36}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({13824}{}{{x}}^{{8}}{+}{42048}{}{{x}}^{{7}}{+}{56832}{}{{x}}^{{6}}{+}{42816}{}{{x}}^{{5}}{+}{15072}{}{{x}}^{{4}}{+}{1068}{}{{x}}^{{3}}{-}{264}{}{{x}}^{{2}}{-}{108}{}{x}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({18432}{}{{x}}^{{7}}{+}{53376}{}{{x}}^{{6}}{+}{71616}{}{{x}}^{{5}}{+}{53952}{}{{x}}^{{4}}{+}{18336}{}{{x}}^{{3}}{+}{1416}{}{{x}}^{{2}}{-}{180}{}{x}{-}{72}\right){}{\mathrm{Dx}}{+}{4608}{}{{x}}^{{6}}{+}{12672}{}{{x}}^{{5}}{+}{16896}{}{{x}}^{{4}}{+}{12672}{}{{x}}^{{3}}{+}{4128}{}{{x}}^{{2}}{+}{360}{}{x}$ (7)
 > $\mathrm{DFactor}\left(L,\left[\mathrm{Dx},x\right]\right)$
 $\left[\left({192}{}{{x}}^{{10}}{+}{640}{}{{x}}^{{9}}{+}{880}{}{{x}}^{{8}}{+}{656}{}{{x}}^{{7}}{+}{244}{}{{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{-}{7}{}{{x}}^{{4}}{-}{3}{}{{x}}^{{3}}\right){}\left({{\mathrm{Dx}}}^{{2}}{+}\frac{{2}{}\left({1152}{}{{x}}^{{7}}{+}{3616}{}{{x}}^{{6}}{+}{4912}{}{{x}}^{{5}}{+}{3696}{}{{x}}^{{4}}{+}{1328}{}{{x}}^{{3}}{+}{90}{}{{x}}^{{2}}{-}{29}{}{x}{-}{12}\right){}{\mathrm{Dx}}}{{x}{}\left({192}{}{{x}}^{{7}}{+}{640}{}{{x}}^{{6}}{+}{880}{}{{x}}^{{5}}{+}{656}{}{{x}}^{{4}}{+}{244}{}{{x}}^{{3}}{+}{16}{}{{x}}^{{2}}{-}{7}{}{x}{-}{3}\right)}{+}\frac{{2}{}\left({2880}{}{{x}}^{{7}}{+}{8480}{}{{x}}^{{6}}{+}{11408}{}{{x}}^{{5}}{+}{8592}{}{{x}}^{{4}}{+}{2956}{}{{x}}^{{3}}{+}{222}{}{{x}}^{{2}}{-}{37}{}{x}{-}{15}\right)}{{{x}}^{{2}}{}\left({192}{}{{x}}^{{7}}{+}{640}{}{{x}}^{{6}}{+}{880}{}{{x}}^{{5}}{+}{656}{}{{x}}^{{4}}{+}{244}{}{{x}}^{{3}}{+}{16}{}{{x}}^{{2}}{-}{7}{}{x}{-}{3}\right)}\right){,}{\mathrm{Dx}}{+}\frac{{2}}{{x}}{,}{\mathrm{Dx}}{+}\frac{{2}}{{x}}\right]$ (8)
 > $\mathrm{map}\left(\mathrm{degree},,\mathrm{Dx}\right)$
 $\left[{2}{,}{1}{,}{1}\right]$ (9)

The first order factors of $L$ come from the integrals. The second order factor comes from the hypergeometric 2F1 function.

Summing special functions preserves the order if they are contiguous (parameter differences are integers).

 > $\mathrm{FindODE}\left(\mathrm{BesselI}\left(0,x\right)+x\mathrm{BesselI}\left(2,x\right),y\left(x\right)\right)$
 $\left({-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{-}{4}{}{{x}}^{{2}}{-}{3}{}{x}\right){}{y}{}\left({x}\right){+}\left({-}{{x}}^{{3}}{+}{x}{+}{4}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{2}{}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (10)

Products of two contiguous second-order special functions satisfy third order equations.

 > $\mathrm{FindODE}\left(\mathrm{BesselI}\left(0,\mathrm{sqrt}\left(x\right)\right)\mathrm{BesselI}\left(3,\mathrm{sqrt}\left(x\right)\right),\left[\mathrm{Dx},x\right]\right)$
 $\left({24}{}{{x}}^{{4}}{+}{64}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({84}{}{{x}}^{{3}}{+}{288}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({-}{24}{}{{x}}^{{3}}{-}{130}{}{{x}}^{{2}}{+}{16}{}{x}\right){}{\mathrm{Dx}}{-}{24}{}{{x}}^{{2}}{-}{179}{}{x}{-}{216}$ (11)
 > $\mathrm{FindODE}\left({\mathrm{hypergeom}\left(\left[\frac{1}{3},\frac{2}{3}\right],\left[1\right],x\right)}^{2}-\mathrm{hypergeom}\left(\left[\frac{1}{3},-\frac{1}{3}\right],\left[1\right],x\right)\mathrm{hypergeom}\left(\left[\frac{4}{3},\frac{2}{3}\right],\left[1\right],x\right),\left[\mathrm{Dx},x\right]\right)$
 $\left({9}{}{{x}}^{{5}}{-}{27}{}{{x}}^{{3}}{+}{18}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({45}{}{{x}}^{{4}}{+}{54}{}{{x}}^{{3}}{-}{135}{}{{x}}^{{2}}{+}{36}{}{x}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({38}{}{{x}}^{{3}}{+}{110}{}{{x}}^{{2}}{-}{94}{}{x}\right){}{\mathrm{Dx}}{+}{2}{}{{x}}^{{2}}{+}{18}{}{x}{-}{2}$ (12)

Non-contiguous examples:

 > $\mathrm{FindODE}\left(\mathrm{KummerM}\left(\frac{1}{4},1,x\right)+\mathrm{KummerM}\left(-\frac{1}{4},1,x\right),\left[\mathrm{Dx},x\right]\right)$
 ${16}{}{{x}}^{{2}}{}{{\mathrm{Dx}}}^{{4}}{+}\left({-}{32}{}{{x}}^{{2}}{+}{64}{}{x}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({16}{}{{x}}^{{2}}{-}{80}{}{x}{+}{32}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({16}{}{x}{-}{16}\right){}{\mathrm{Dx}}{-}{1}$ (13)
 > $\mathrm{FindODE}\left(\mathrm{LegendreP}\left(\frac{1}{4},x\right)\mathrm{LegendreP}\left(\frac{1}{2},x\right),y\left(x\right)\right)$
 ${-}{495}{}{y}{}\left({x}\right){+}{1440}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({5600}{}{{x}}^{{2}}{-}{1504}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({2560}{}{{x}}^{{3}}{-}{2560}{}{x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({256}{}{{x}}^{{4}}{-}{512}{}{{x}}^{{2}}{+}{256}\right){}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (14)
 > $\mathrm{dsolve}\left(\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{LegendreP}}{}\left(\frac{{1}}{{2}}{,}{x}\right){}{\mathrm{LegendreQ}}{}\left(\frac{{1}}{{4}}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{LegendreP}}{}\left(\frac{{1}}{{4}}{,}{x}\right){}{\mathrm{LegendreP}}{}\left(\frac{{1}}{{2}}{,}{x}\right){+}{\mathrm{_C3}}{}{\mathrm{LegendreQ}}{}\left(\frac{{1}}{{2}}{,}{x}\right){}{\mathrm{LegendreP}}{}\left(\frac{{1}}{{4}}{,}{x}\right){+}{\mathrm{_C4}}{}{\mathrm{LegendreQ}}{}\left(\frac{{1}}{{2}}{,}{x}\right){}{\mathrm{LegendreQ}}{}\left(\frac{{1}}{{4}}{,}{x}\right)$ (15)

Compatibility

 • The DEtools[FindODE] command was introduced in Maple 2019.