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.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{FindODE}\left(\cos \left(\mathrm{sqrt}\left(x\right)\right)\,y\left(x\right)\right)$

$y\left(x\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)+4x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{FindODE}\left(\exp \left(x\right)+\mathrm{sqrt}\left(x\right)\,y\left(x\right)\right)$

$\left(2x+1\right)y\left(x\right)+\left(-4x^2-1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(4x^2-2x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{FindODE}\left(\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)\,y\left(x\right)\right)$

$aby\left(x\right)+\left(xa+xb-c+x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

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

$\mathrm{FAIL}$

$\mathrm{FindODE}\left(\cos \left(\mathrm{sqrt}\left(x\right)\right)\,\left[\mathrm{Dx}\,x\right]\right)$

$4x{\mathrm{Dx}}^2+2\mathrm{Dx}+1$

$\mathrm{ogf}≔x^{-2}\left(\mathrm{Int}\left(\mathrm{Int}\left(\frac{2\mathrm{hypergeom}\left(\left[\frac{3}{4}\,\frac{5}{4}\right]\,\left[2\right]\,\frac{64x^3\left(1+x\right)}{{\left(1-4x^2\right)}^2}\right)}{{\left(1-4x^2\right)}^{\frac{3}{2}}}\,x\right)\,x\right)\right)$

$\mathrm{ogf}≔\frac{\int \int \frac{2\mathrm{hypergeom}\left(\left[\frac{3}{4}\,\frac{5}{4}\right]\,\left[2\right]\,\frac{64x^3\left(x+1\right)}{{\left(-4x^2+1\right)}^2}\right)}{{\left(-4x^2+1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\ⅆx\ⅆx}{x^2}$

$L≔\mathrm{FindODE}\left(\mathrm{ogf}\,\left[\mathrm{Dx}\,x\right]\right)$

$L≔\left(192x^{10}+640x^9+880x^8+656x^7+244x^6+16x^5-7x^4-3x^3\right){\mathrm{Dx}}^4+\left(3072x^9+9792x^8+13344x^7+10016x^6+3632x^5+244x^4-86x^3-36x^2\right){\mathrm{Dx}}^3+\left(13824x^8+42048x^7+56832x^6+42816x^5+15072x^4+1068x^3-264x^2-108x\right){\mathrm{Dx}}^2+\left(18432x^7+53376x^6+71616x^5+53952x^4+18336x^3+1416x^2-180x-72\right)\mathrm{Dx}+4608x^6+12672x^5+16896x^4+12672x^3+4128x^2+360x$

$\mathrm{DFactor}\left(L\,\left[\mathrm{Dx}\,x\right]\right)$

$\left[\left(192x^{10}+640x^9+880x^8+656x^7+244x^6+16x^5-7x^4-3x^3\right)\left({\mathrm{Dx}}^2+\frac{2\left(1152x^7+3616x^6+4912x^5+3696x^4+1328x^3+90x^2-29x-12\right)\mathrm{Dx}}{x\left(192x^7+640x^6+880x^5+656x^4+244x^3+16x^2-7x-3\right)}+\frac{2\left(2880x^7+8480x^6+11408x^5+8592x^4+2956x^3+222x^2-37x-15\right)}{x^2\left(192x^7+640x^6+880x^5+656x^4+244x^3+16x^2-7x-3\right)}\right)\,\mathrm{Dx}+\frac{2}{x}\,\mathrm{Dx}+\frac{2}{x}\right]$

$\mathrm{map}\left(\mathrm{degree}\,\,\mathrm{Dx}\right)$

$\left[2\,1\,1\right]$

$\mathrm{FindODE}\left(\mathrm{BesselI}\left(0\,x\right)+x\mathrm{BesselI}\left(2\,x\right)\,y\left(x\right)\right)$

$\left(-x^4-2x^3-4x^2-3x\right)y\left(x\right)+\left(-x^3+x+4\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^4+2x^3+x^2+2x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{FindODE}\left(\mathrm{BesselI}\left(0\,\mathrm{sqrt}\left(x\right)\right)\mathrm{BesselI}\left(3\,\mathrm{sqrt}\left(x\right)\right)\,\left[\mathrm{Dx}\,x\right]\right)$

$\left(24x^4+64x^3\right){\mathrm{Dx}}^3+\left(84x^3+288x^2\right){\mathrm{Dx}}^2+\left(-24x^3-130x^2+16x\right)\mathrm{Dx}-24x^2-179x-216$

$\mathrm{FindODE}\left({\mathrm{hypergeom}\left(\left[\frac{1}{3}\,\frac{2}{3}\right]\,\left[1\right]\,x\right)}^2-\mathrm{hypergeom}\left(\left[\frac{1}{3}\,-\frac{1}{3}\right]\,\left[1\right]\,x\right)\mathrm{hypergeom}\left(\left[\frac{4}{3}\,\frac{2}{3}\right]\,\left[1\right]\,x\right)\,\left[\mathrm{Dx}\,x\right]\right)$

$\left(9x^5-27x^3+18x^2\right){\mathrm{Dx}}^3+\left(45x^4+54x^3-135x^2+36x\right){\mathrm{Dx}}^2+\left(38x^3+110x^2-94x\right)\mathrm{Dx}+2x^2+18x-2$

$\mathrm{FindODE}\left(\mathrm{KummerM}\left(\frac{1}{4}\,1\,x\right)+\mathrm{KummerM}\left(-\frac{1}{4}\,1\,x\right)\,\left[\mathrm{Dx}\,x\right]\right)$

$16x^2{\mathrm{Dx}}^4+\left(-32x^2+64x\right){\mathrm{Dx}}^3+\left(16x^2-80x+32\right){\mathrm{Dx}}^2+\left(16x-16\right)\mathrm{Dx}-1$

$\mathrm{FindODE}\left(\mathrm{LegendreP}\left(\frac{1}{4}\,x\right)\mathrm{LegendreP}\left(\frac{1}{2}\,x\right)\,y\left(x\right)\right)$

$-495y\left(x\right)+1440x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(5600x^2-1504\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(2560x^3-2560x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+\left(256x^4-512x^2+256\right)\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)$

$\mathrm{dsolve}\left(\right)$

$y\left(x\right)=\mathrm{\_C1}\mathrm{LegendreP}\left(\frac{1}{2}\,x\right)\mathrm{LegendreQ}\left(\frac{1}{4}\,x\right)+\mathrm{\_C2}\mathrm{LegendreP}\left(\frac{1}{4}\,x\right)\mathrm{LegendreP}\left(\frac{1}{2}\,x\right)+\mathrm{\_C3}\mathrm{LegendreQ}\left(\frac{1}{2}\,x\right)\mathrm{LegendreP}\left(\frac{1}{4}\,x\right)+\mathrm{\_C4}\mathrm{LegendreQ}\left(\frac{1}{2}\,x\right)\mathrm{LegendreQ}\left(\frac{1}{4}\,x\right)$

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

For more information on Maple 2019 changes, see Updates in Maple 2019.