Given an integrating factor (x,y,..._yn), where $\mathrm{\_yn}$ represents the nth order derivative, or a list of them, redode returns the ODE of order n having all the values of mu as integrating factors.

This command is useful to identify the general ODE problem related to a given mu, and to understand the possible links between the integrating factor scheme for reducing the order and other reduction schemes (for example, symmetries).

When the expected reduced ODE, here called R, is also given as argument, its differential order, say $m$, must satisfy $m<n$. The routine then proceeds as follows. First, it redefines R by differentiating it such that its differential order becomes $n-1$. Then it performs a test to see if the problem is solvable. The test for solvability is

mu(x,y,`...`,_ym) = nu(x,y,`...`,_y||(m-1))*diff(R(x,y,`...`,_ym),_ym);

$\mathrm{\mu }\left(x\&comma;y\&comma;\mathrm{...}\&comma;\mathrm{\_ym}\right)=\mathrm{\nu }\left(x\&comma;y\&comma;\mathrm{...}\&comma;\mathrm{\_y}\Vert \left(m-1\right)\right)\left(\frac{\partial }{\partial \mathrm{\_ym}}R\left(x\&comma;y\&comma;\mathrm{...}\&comma;\mathrm{\_ym}\right)\right)$

for some function $\mathrm{nu}\left(x,y,...,\mathrm{\_ym}\right)$. If the problem is solvable, redode then returns an nth order ODE satisfying

mu(x,y,`...`,diff(y(x),x$m)) * ODE=Diff(R(x,y(x),`...`,diff(y(x),x$m)),x);

$\mathrm{\mu }\left(x\&comma;y\&comma;\mathrm{...}\&comma;\frac{{\&DifferentialD;}^m}{\&DifferentialD;x^m}y\left(x\right)\right)\mathrm{ODE}=\frac{\partial }{\partial x}R\left(x\&comma;y\left(x\right)\&comma;\mathrm{...}\&comma;\frac{{\&DifferentialD;}^m}{\&DifferentialD;x^m}y\left(x\right)\right)$

Note that the reduced ODE being differentiated on the right-hand side is defined up to a constant. For example, $R+\mathrm{\_C1}$ also satisfies the above equation.

When the given mu does not depend on $\mathrm{\_ym}$ and R is nonlinear in $\mathrm{\_ym}$, the requested nth order ODE exists if R can be solved for $\mathrm{\_ym}$; then nu can be determined as the ratio $\frac{\mathrm{\mu }\mathrm{d\_ym}}{\mathrm{dR}}$.

This function is part of the DEtools package, and so it can be used in the form redode(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[redode](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\&colon;$

$\mathrm{ode\_psi}≔\mathrm{diff}\left(y\left(x\right)\&comma;x\&comma;x\right)=\mathrm{\&psi;1}\left(x\right)\mathrm{diff}\left(y\left(x\right)\&comma;x\right)+\mathrm{\&psi;2}\left(x\right)y\left(x\right)+\mathrm{\&psi;3}\left(x\right)$

$\mathrm{ode\_psi}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=\mathrm{\&psi;1}\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)+\mathrm{\&psi;2}\left(x\right)y\left(x\right)+\mathrm{\&psi;3}\left(x\right)$

$\mathrm{ode\_1}≔\mathrm{redode}\left(F\left(x\right)\&comma;y\left(x\right)\right)$

$\mathrm{ode\_1}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=-\frac{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)y\left(x\right)+\mathrm{\_F1}\left(x\right)}{F\left(x\right)}$

$\mathrm{ode\_2}≔\mathrm{redode}\left(F\left(x\right)\&comma;y\left(x\right)\&comma;\mathrm{ode\_1}\right)$

$\mathrm{ode\_2}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=-\frac{2\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)+\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}F\left(x\right)\right)y\left(x\right)+\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{\_F1}\left(x\right)}{F\left(x\right)}$

$\mathrm{e1}≔-\frac{2}{F\left(x\right)}\mathrm{diff}\left(F\left(x\right)\&comma;x\right)=\mathrm{\&psi;1}\left(x\right)$

$\mathrm{e1}≔-\frac{2\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)}{F\left(x\right)}=\mathrm{\&psi;1}\left(x\right)$

$\mathrm{e2}≔-\frac{1}{F\left(x\right)}\mathrm{diff}\left(F\left(x\right)\&comma;x\&comma;x\right)=\mathrm{\&psi;2}\left(x\right)$

$\mathrm{e2}≔-\frac{\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}F\left(x\right)}{F\left(x\right)}=\mathrm{\&psi;2}\left(x\right)$

$\mathrm{ans\_F}≔\mathrm{dsolve}\left(\mathrm{e1}\right)$

$\mathrm{ans\_F}≔F\left(x\right)=\mathrm{c\_\_1}{\&ExponentialE;}^{\int -\frac{\mathrm{\&psi;1}\left(x\right)}{2}\&DifferentialD;x}$

$\mathrm{ode\_pattern}≔\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{e2}\&comma;\mathrm{ans\_F}\right)\right)$

$\mathrm{ode\_pattern}≔\frac{\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{\&psi;1}\left(x\right)}{2}-\frac{{\mathrm{\&psi;1}\left(x\right)}^2}{4}=\mathrm{\&psi;2}\left(x\right)$

$\mathrm{ode\_1}≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)$

$\mathrm{ode\_1}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)$

$\mathrm{ode\_2}≔\mathrm{redode}\left(F\left(x\right)\&comma;y\left(x\right)\&comma;\mathrm{ode\_1}\right)$

$\mathrm{ode\_2}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=\frac{A\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)y\left(x\right)+A\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)F\left(x\right)+\left(\frac{\&DifferentialD;}{\&DifferentialD;x}A\left(x\right)\right)y\left(x\right)F\left(x\right)+B\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)+\left(\frac{\&DifferentialD;}{\&DifferentialD;x}B\left(x\right)\right)F\left(x\right)-\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}{F\left(x\right)}$

$\mathrm{constraint}≔\mathrm{expand}\left(\mathrm{odepde}\left(\mathrm{ode\_2}\&comma;y\left(x\right)\&comma;\left[0\&comma;F\left(x\right)\right]\right)\right)$

$\mathrm{constraint}≔-2A\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)-\left(\frac{\&DifferentialD;}{\&DifferentialD;x}A\left(x\right)\right)F\left(x\right)+\frac{{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)}^2}{F\left(x\right)}+\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}F\left(x\right)$

$\mathrm{ans\_A}≔\mathrm{dsolve}\left(\mathrm{constraint}\&comma;A\left(x\right)\right)$

$\mathrm{ans\_A}≔A\left(x\right)=\frac{F\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)+\mathrm{c\_\_1}}{{F\left(x\right)}^2}$

$\mathrm{ode\_2\_H}≔\mathrm{diff}\left(y\left(x\right)\&comma;x\&comma;x\right)=\mathrm{collect}\left(\mathrm{select}\left(\mathrm{has}\&comma;\mathrm{expand}\left(\mathrm{rhs}\left(\mathrm{ode\_2}\right)\right)\&comma;y\right)\&comma;\left\{\mathrm{diff}\left(y\left(x\right)\&comma;x\right)\&comma;y\left(x\right)\right\}\right)$

$\mathrm{ode\_2\_H}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=\left(A\left(x\right)-\frac{\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)}{F\left(x\right)}\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)+\left(\frac{A\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)}{F\left(x\right)}+\frac{\&DifferentialD;}{\&DifferentialD;x}A\left(x\right)\right)y\left(x\right)$

$\mathrm{ode\_pattern}≔\mathrm{subs}\left(\mathrm{\_C1}=1\&comma;\mathrm{collect}\left(\mathrm{expand}\left(\mathrm{subs}\left(\mathrm{ans\_A}\&comma;\mathrm{ode\_2\_H}\right)\right)\&comma;y\right)\right)$

$\mathrm{ode\_pattern}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=\left(-\frac{\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)}{{F\left(x\right)}^3}+\frac{\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}F\left(x\right)}{F\left(x\right)}\right)y\left(x\right)+\frac{\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)}{{F\left(x\right)}^2}$

$\mathrm{\mu }≔\left[y\left(x\right)x\&comma;x^{-\frac{1}{4}+\frac{1}{4}{29}^{\frac{1}{2}}}y\left(x\right)\&comma;x^{-\frac{1}{4}-\frac{1}{4}{29}^{\frac{1}{2}}}y\left(x\right)\right]$

$\mathrm{\mu }≔\left[y\left(x\right)x\&comma;x^{-\frac{1}{4}+\frac{\sqrt{29}}{4}}y\left(x\right)\&comma;x^{-\frac{1}{4}-\frac{\sqrt{29}}{4}}y\left(x\right)\right]$

$\mathrm{redode}\left(\mathrm{\mu }\&comma;3\&comma;y\left(x\right)\right)$

$\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}y\left(x\right)=\frac{13y\left(x\right)}{8x^3}+\frac{5\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right)}{2x}-\frac{13\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}{4x^2}+\frac{-3\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right)+\frac{5{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}^2}{2x}+\mathrm{f\_\_1}\left(x\right)}{y\left(x\right)}$

$\mathrm{map}\left(\mathrm{DEtools}\left[\mathrm{mutest}\right]\&comma;\mathrm{\mu }\&comma;\right)$

$\left[0\&comma;0\&comma;0\right]$

$\mathrm{\mu }\left[1..2\right]$

$\left[y\left(x\right)x\&comma;x^{-\frac{1}{4}+\frac{\sqrt{29}}{4}}y\left(x\right)\right]$

$\mathrm{redode}\left(\&comma;3\&comma;y\left(x\right)\right)$

$\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}y\left(x\right)={\mathrm{D}}_2\left(\mathrm{f\_\_3}\right)\left(x\&comma;y\left(x\right)\right){\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}^2+\frac{\left(16{\mathrm{D}}_1\left(\mathrm{f\_\_3}\right)\left(x\&comma;y\left(x\right)\right)x^2+2x\mathrm{f\_\_3}\left(x\&comma;y\left(x\right)\right)\sqrt{29}-2x\mathrm{f\_\_3}\left(x\&comma;y\left(x\right)\right)-5\sqrt{29}+19\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}{8x^2}+\mathrm{f\_\_3}\left(x\&comma;y\left(x\right)\right)\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right)+\frac{\mathrm{f\_\_3}\left(x\&comma;y\left(x\right)\right){\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)}^2-3\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right)+{\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{\left(2\sqrt{29}\left(\frac{\partial }{\partial x}\mathrm{f\_\_3}\left(x\&comma;\mathrm{\_a}\right)\right)x^2+8\left(\frac{{\partial }^2}{\partial x^2}\mathrm{f\_\_3}\left(x\&comma;\mathrm{\_a}\right)\right)x^3-2\left(\frac{\partial }{\partial x}\mathrm{f\_\_3}\left(x\&comma;\mathrm{\_a}\right)\right)x^2+5\sqrt{29}-19\right)\mathrm{\_a}}{8x^3}\&DifferentialD;\mathrm{\_a}+\mathrm{f\_\_4}\left(x\right)}{y\left(x\right)}$

$\mathrm{map}\left(\mathrm{DEtools}\left[\mathrm{mutest}\right]\&comma;\mathrm{\mu }\left[1..2\right]\&comma;\right)$

$\left[0\&comma;0\right]$