The power_equivalent and rational_equivalent commands return an ODE (equivalent to the given ODE), with rational coefficients, and the transformation that maps the returned ODE back to the given ODE. The returned ODE has some additional relevant features.

It is in normal form.

All its invariants are rational.

The degree of each invariant with respect to the independent variable is the lowest one achievable using power (power_equivalent) or rational (rational_equivalent) transformations.

The use of a NormalForm where in addition item 3 holds makes it possible to solve equivalence problems under compositions of Mobius and power transformations between a given ODE and ODEs admitting, for instance, hypergeometric solutions (see dsolve/hyper3).

The transformation used to compute the equivalent ODE composes arbitrary powers (for power_equivalent) or rational functions (for rational_equivalent) of the independent variable with the most general linear transformation of the dependent variable, so they are of the form $\left\{x=t^a\,y\left(x\right)=P\left(t\right)u\left(t\right)+Q\left(t\right)\right\}$. In the case of linear homogeneous ODEs, $Q=0$. In terms of linear ODEs, the rational_equivalent command only handles those with order > 2. The transformations preserve the polynomial degree, the linear or nonlinear type and the homogeneous character in the case of linear equations. For example, second order linear equations are mapped into equations of the same type, the same way as Abel or Riccati type nonlinear equations are mapped into equations of the same type.

By comparing the transformations used by power_equivalent and rational_equivalent with the general transformation $\left\{x=F\left(t\right)\,y\left(x\right)=P\left(t\right)u\left(t\right)+Q\left(t\right)\right\}$ which defines the equivalence class associated to any polynomial ODE (linear or not), you can see that the restrictions imposed by power_equivalent and rational_equivalent respectively consist of taking $F\left(t\right)=t^a$ and F(t)=rational function of t. Due to these restrictions, for some given ODE, the equivalence with a rational coefficients ODE may not exist. However, if in the given ODE, when written in normal form (see convert/NormalForm), all the nonrational objects are either fractional or abstract powers of the independent variable, then the equivalence problem has a solution, which the power_equivalent command should find. Caveat: Despite the theory behind the algorithm, in the case of abstract powers, the success relies on the ability of the simplification routines to rewrite these powers in a normal form. (This may be a difficult problem; see evala/Normal.)

When the equivalence does not exist power_equivalent and rational_equivalent return NULL. Otherwise, the output sequence consists of a rational ODE and the transformation mapping it into the given ODE. New variables of global type are introduced in the result.

Optionally, you can specify that new dependent $u\left(t\right)$ and independent $t$ variables be used in the output by including the option $u\left(t\right)$ in the calling sequence.

By using the keep option, you specify that the returned rational ODE should be expressed using the original variables $\left\{x\,y\left(x\right)\right\}$. In that case, the returned transformation maps the returned rational ODE into the given ODE expressed using the new variables $\left\{t\,u\left(t\right)\right\}$.

Note: The keep option supersedes the u(t) parameter. If superseded by the keep parameter, $\left\{t\,u\left(t\right)\right\}$ represents the original variables in the transformation back to the given ODE. (See the examples.)

Parts of the code for the command rational_equivalent were written by Austin Roche.

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

$\mathrm{diff}\left(y\left(x\right)\,x\right)+f\left(x\right)y\left(x\right)=0$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)=0$

$\mathrm{power\_equivalent}\left(\right)$

$\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=0,\left\{\mathrm{\_a}=x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=y\left(x\right){\ⅇ}^{-\left(\int -f\left(x\right)\ⅆx\right)}\right\}$

$\mathrm{factor}\left(\mathrm{PDEtools}\left[\mathrm{dchange}\right]\left(\left[2\right]\,\left[1\right]\,\left[x\,y\left(x\right)\right]\right)\right)$

${\ⅇ}^{-\left(\int -f\left(x\right)\ⅆx\right)}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)\right)=0$

$\mathrm{non\_rational\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{\frac{1}{4}\left(\frac{\frac{2}{3}\cdot 1}{x^{\frac{5}{6}}}+\frac{2}{x^{\frac{2}{3}}}\right)}{x}y\left(x\right)+\frac{\frac{1}{2}\left(-1+x^{\frac{1}{6}}\right)}{x}\mathrm{diff}\left(y\left(x\right)\,x\right)$

$\mathrm{non\_rational\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\left(\frac{2}{3x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\frac{2}{x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y\left(x\right)}{4x}+\frac{\left(-1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{2x}$

$\mathrm{rational\_ode},\mathrm{tr}≔\mathrm{power\_equivalent}\left(\mathrm{non\_rational\_ode}\,u\left(t\right)\right)$

$\mathrm{rational\_ode},\mathrm{tr}≔\frac{{\ⅆ}^2}{\ⅆt^2}u\left(t\right)=\left(\frac{81}{4}+\frac{9}{t}+\frac{2}{t^2}\right)u\left(t\right),\left\{t=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,u\left(t\right)=\frac{y\left(x\right){\ⅇ}^{-\left(\int \frac{-1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{4x}\ⅆx\right)}}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}}\right\}$

$\mathrm{sol\_rational}≔\mathrm{dsolve}\left(\mathrm{rational\_ode}\right)$

$\mathrm{sol\_rational}≔u\left(t\right)=\frac{\mathrm{c\_\_1}{\ⅇ}^{-\frac{9t}{2}}}{t}+\frac{\mathrm{c\_\_2}{\ⅇ}^{\frac{9t}{2}}\left(81t^2-18t+2\right)}{t}$

$\mathrm{sol\_non\_rational}≔\mathrm{eval}\left(\mathrm{sol\_rational}\,\mathrm{tr}\right)$

$\mathrm{sol\_non\_rational}≔\frac{y\left(x\right){\ⅇ}^{-\left(\int \frac{-1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{4x}\ⅆx\right)}}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}}=\frac{\mathrm{c\_\_1}{\ⅇ}^{-\frac{9x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{2}}}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\frac{\mathrm{c\_\_2}{\ⅇ}^{\frac{9x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{2}}\left(81x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-18x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}+2\right)}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$

$\mathrm{simplify}\left(\mathrm{isolate}\left(\,y\left(x\right)\right)\right)$

$y\left(x\right)={\ⅇ}^{-\frac{9x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{2}+\frac{\left(\int \frac{-1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{x}\ⅆx\right)}{4}}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(81\mathrm{c\_\_2}\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{9}+\frac{2}{81}\right){\ⅇ}^{9x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\mathrm{c\_\_1}\right)$

$\mathrm{odetest}\left(\,\mathrm{non\_rational\_ode}\right)$

$0$

$\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{\frac{1}{4}\left(x^{-\frac{1}{2}-\frac{1}{2}n}n+2x^{-n}\right)}{x}y\left(x\right)+\frac{\frac{1}{2}\left(-1+x^{-\frac{1}{2}n+\frac{1}{2}}\right)}{x}\mathrm{diff}\left(y\left(x\right)\,x\right)$

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

$\mathrm{power\_equivalent}\left(\,u\left(t\right)\,\mathrm{keep}\right)$

$\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(\frac{9}{4{\left(n-1\right)}^2}+\frac{3n}{2{\left(n-1\right)}^{2}x}-\frac{n^2-2n}{4{\left(n-1\right)}^2{x}^2}\right)y\left(x\right),\left\{x=t^{-\frac{n}{2}+\frac{1}{2}}\,y\left(x\right)=t^{-\frac{1}{4}-\frac{n}{4}}u\left(t\right){\ⅇ}^{-\left(\int \frac{-1+t^{-\frac{n}{2}+\frac{1}{2}}}{4t}\ⅆt\right)}\right\}$

$\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,5\right)\right)=-\frac{5}{x}\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,4\right)\right)-\frac{\frac{15}{4}\cdot 1}{x^2}\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)+\frac{\frac{1}{32}ay\left(x\right)}{x^{\frac{15}{2}}}$

$\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=-\frac{5\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)}{x}-\frac{15\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{4x^2}+\frac{ay\left(x\right)}{32x^{\raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{power\_equivalent}\left(\,u\left(t\right)\,\mathrm{keep}\right)$

$\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=\left(\frac{19008}{625x^5}+\frac{a}{3125x^6}\right)y\left(x\right)-\frac{18504\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{625x^4}+\frac{72\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{5x^3}-\frac{24\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{5x^2},\left\{x=t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\,y\left(x\right)=t^{3}u\left(t\right){\ⅇ}^{-\left(\int -\frac{1}{t}\ⅆt\right)}\right\}$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,5\right)\right)=xy\left(x\right)$

$\mathrm{ode}≔\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=xy\left(x\right)$

$\mathrm{convert}\left(\mathrm{ode}\,\mathrm{NormalForm}\,\mathrm{keep}\right)$

$\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=xy\left(x\right),\left\{\mathrm{\_a}\left(x\right)=\mathrm{c\_\_1}y\left(x\right)\right\}$

$\mathrm{power\_equivalent}\left(\mathrm{ode}\,\mathrm{keep}\right)$

$\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=\left(\frac{1}{7776x^4}+\frac{5005}{162x^5}\right)y\left(x\right)-\frac{4865\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{162x^4}+\frac{175\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{12x^3}-\frac{175\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{36x^2},\left\{x={\mathrm{\_a}}^6\,y\left(x\right)={\mathrm{\_a}}^{10}\mathrm{\_b}\left(\mathrm{\_a}\right)\right\}$

$\mathrm{diff}\left(y\left(x\right)\,x\right)=-\frac{\frac{3}{2}\left(x^{\frac{1}{2}}+1\right)\left(x-x^{\frac{1}{2}}+1\right)}{x^{\frac{11}{2}}}{y\left(x\right)}^3-\frac{15}{4x^{\frac{13}{4}}}{y\left(x\right)}^2$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{3\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right){y\left(x\right)}^3}{2x^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{15{y\left(x\right)}^2}{4x^{\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

$\mathrm{power\_equivalent}\left(\right)$

$\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=1-\frac{25{\left(2+9\mathrm{\_a}\right)}^2{\mathrm{\_b}\left(\mathrm{\_a}\right)}^3}{11664{\mathrm{\_a}}^3{\left(\mathrm{\_a}+1\right)}^3}+\frac{\left(54{\mathrm{\_a}}^2+99\mathrm{\_a}+38\right)\mathrm{\_b}\left(\mathrm{\_a}\right)}{108{\mathrm{\_a}}^3+132{\mathrm{\_a}}^2+24\mathrm{\_a}},\left\{\mathrm{\_a}=x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\,\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{18\left(6y\left(x\right)x^3+12y\left(x\right)x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+5x^{\raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+5x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+6y\left(x\right)\right)}{5x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(9x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+2\right)}\right\}$

$\mathrm{diff}\left(y\left(x\right)\,x\right)=1+\frac{12x^{\frac{7}{2}}+a}{x^{\frac{5}{2}}y\left(x\right)}$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=1+\frac{12x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+a}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}y\left(x\right)}$

$\mathrm{power\_equivalent}\left(\right)$

$\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=1-\frac{50{\left(44\mathrm{\_a}-9a\right)}^2{\mathrm{\_b}\left(\mathrm{\_a}\right)}^3}{250047{\mathrm{\_a}}^2{\left(12\mathrm{\_a}+a\right)}^3}+\frac{2\left(-7172{\mathrm{\_a}}^2+2931\mathrm{\_a}a+27a^2\right)\mathrm{\_b}\left(\mathrm{\_a}\right)}{21\mathrm{\_a}\left(-44\mathrm{\_a}+9a\right)\left(12\mathrm{\_a}+a\right)},\left\{\mathrm{\_a}=x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\,\mathrm{\_b}\left(\mathrm{\_a}\right)=-\frac{63x\left(12y\left(x\right)x^6+432x^7+y\left(x\right)x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}a+72x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}a+3a^2\right)}{5y\left(x\right)\left(44x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-9a\right)}\right\}$

$\mathrm{diff}\left(y\left(x\right)\,x\right)=-\frac{6x^5+A}{x^4}{y\left(x\right)}^3-{y\left(x\right)}^2$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{\left(6x^5+A\right){y\left(x\right)}^3}{x^4}-{y\left(x\right)}^2$

$\mathrm{convert}\left(\,\mathrm{Abel\_RNF}\,\mathrm{keep}\right)$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=1-\frac{16x^2{\left(-14x^5+9A\right)}^2{y\left(x\right)}^3}{729{\left(6x^5+A\right)}^3}+\frac{\left(-518x^{10}+1479A{x}^5-81A^2\right)y\left(x\right)}{27x\left(-\frac{14x^5}{9}+A\right)\left(6x^5+A\right)},\left\{\mathrm{\_a}\left(x\right)=\frac{x^3\left(\left(-56x^5+36A\right)y\left(x\right)-54x^6-9Ax\right)}{27{\left(6x^5+A\right)}^2}\right\}$

$\mathrm{power\_equivalent}\left(\,\mathrm{keep}\right)$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{16{\left(-14x+9A\right)}^2{y\left(x\right)}^3}{91125x^2{\left(6x+A\right)}^3}+\frac{\left(27A^2+1959Ax-1526x^2\right)y\left(x\right)}{15x\left(-14x+9A\right)\left(6x+A\right)}+1,\left\{x={\mathrm{\_a}}^5\,y\left(x\right)=\frac{45\mathrm{\_a}\left(108{\mathrm{\_a}}^{10}\mathrm{\_b}\left(\mathrm{\_a}\right)+6{\mathrm{\_a}}^9+36A{\mathrm{\_a}}^5\mathrm{\_b}\left(\mathrm{\_a}\right)+{\mathrm{\_a}}^{4}A+3A^2\mathrm{\_b}\left(\mathrm{\_a}\right)\right)}{4\left(-14{\mathrm{\_a}}^5+9A\right)}\right\}$

$\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{{y\left(x\right)}^2+x^5}{x^5}$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{{y\left(x\right)}^2+x^5}{x^5}$

$\mathrm{power\_equivalent}\left(\,u\left(t\right)\right)$

$\frac{\ⅆ}{\ⅆt}u\left(t\right)={u\left(t\right)}^2-\frac{7t-4}{36t^3},\left\{t=x^3\,u\left(t\right)=\frac{y\left(x\right)}{3x^7}-\frac{7}{6x^3}\right\}$

$\mathrm{PDEtools}:-\mathrm{dchange}\left(\left[2\right]\,\left[1\right]\,\mathrm{normal}\right)$

$\frac{21x^4+2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x-14y\left(x\right)}{18x^{10}}=\frac{21x^8-14y\left(x\right)x^4+2x^5+2{y\left(x\right)}^2}{18x^{14}}$

$\mathrm{normal}\left(\mathrm{isolate}\left(\,\mathrm{diff}\left(y\left(x\right)\,x\right)\right)\right)$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{{y\left(x\right)}^2+x^5}{x^5}$