The reduce_order function uses ODE symmetries or ODE solutions to compute appropriate changes of variables that reduce the order of the input ODE. By default, the output is a solution expressed in terms of the reduced ODE (see ODESolStruc). As with all ODESolStruc, these solutions can be converted to standard closed form solutions by using the DEtools[buildsol] command if you can solve the reduced ODE.

Optionally, in the case of a linear ODE, the solution can also be expressed in terms of DESol. There is an equivalence between DESol and ODESolStruc; the output can also be the reduced ODE itself, or a solution in standard closed form. For these purposes use the optional argument output=<keyword>, where <keyword> is one of: ODESolStruc, DESol, basis, reduced_ode, and solution.

Up to $r$ reductions of order are returned when $r$ symmetries are given and form a solvable Lie Algebra (see solve_group).

Similarly, up to $r$ reductions of order are returned when $r$ independent solutions y=solution1(x), ..., y=solutionr(x) (instead of symmetries) are given.

In the case of a linear ODE, any solution $S\left(x\right)$ is also a symmetry of the form $\left[\mathrm{\xi }=0\&comma;\mathrm{\eta }=S\left(x\right)\right]$; so it is possible to use the symmetry routines with linear ODEs as well. In that case, it suffices to pass the r solutions [S1(x), ..., Sr(x)] written as symmetries [0, S1(x)], ..., [0, Sr(x)].

When many reductions of order are performed, reduce_order can also output in two different manners. The default output is one object (the solution or the reduced ODE) without giving any information on the intermediate reductions of order. By giving the optional argument in_sequence, a Maple sequence of reductions of order is returned such that the first element in the sequence shows the input ODE reduced in order by one, the second element shows the first reduced ODE reduced in order by one, and so on.

When no output=<keyword> is indicated, this sequence is a sequence of ODESolStruc, where each ODESolStruc contains the reduced ODE, the forward transformation of variables from the original ODE to the new ODE, and the reverse transformation. All these transformations can be combined using buildsol.

Optionally, you can indicate the name of the dependent variable new_vars to be used in the reduced ODE or also in each of the reduced ODEs when the in_sequence optional argument is given. To indicate these dependent variables it suffices to specify them in the calling sequence anywhere after the first argument (the ODE to reduce).

If many symmetries or solutions are given, for nonlinear ODEs, reduce_order optimizes the order in which the symmetries or solutions are used. To enforce the use of these symmetries in exactly the given order use the optional argument keep_order.

Two algorithms are used to reduce the order of ODEs using symmetries: the method of differential invariants and the method of canonical coordinates. They can be specified by passing dif or can, respectively, as an extra argument. The default is can. See invariants for more information.

Optional arguments can be specified in any order after the first argument (the ODE to reduce).

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

$\mathrm{sym}≔\left[1\&comma;0\right]$

$\mathrm{sym}≔\left[1\&comma;0\right]$

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

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

$\mathrm{invariants}\left(\mathrm{sym}\&comma;y\left(x\right)\&comma;\mathrm{can}\right)$

$\mathrm{\_I0}=y,\mathrm{\_I1}=\frac{1}{\mathrm{\_y1}}$

$\mathrm{reduce\_order}\left(\mathrm{ode}\&comma;\mathrm{sym}\&comma;u\left(t\right)\&comma;\mathrm{can}\right)$

$y\left(x\right)=t\mathbf{where}\left[\left\{u\left(t\right)=\frac{1}{t}\right\}\&comma;\left\{t=y\left(x\right)\&comma;u\left(t\right)=\frac{1}{\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)}\right\}\&comma;\left\{x=\int u\left(t\right)\&DifferentialD;t+\mathrm{\_C1}\&comma;y\left(x\right)=t\right\}\right]$

$\mathrm{invariants}\left(\mathrm{sym}\&comma;y\left(x\right)\&comma;\mathrm{dif}\right)$

$\mathrm{\_I0}=y,\mathrm{\_I1}=\mathrm{\_y1}$

$\mathrm{reduce\_order}\left(\mathrm{ode}\&comma;\mathrm{sym}\&comma;u\left(t\right)\&comma;\mathrm{dif}\right)$

$y\left(x\right)=t\mathbf{where}\left[\left\{u\left(t\right)=t\right\}\&comma;\left\{t=y\left(x\right)\&comma;u\left(t\right)=\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right\}\&comma;\left\{x=\int \frac{1}{u\left(t\right)}\&DifferentialD;t+\mathrm{\_C1}\&comma;y\left(x\right)=t\right\}\right]$

$\mathrm{op}\left(\left[2\&comma;1\&comma;1\right]\&comma;\mathrm{rhs}\left(\right)\right)$

$u\left(t\right)=t$

$\mathrm{buildsol}\left(\&comma;\right)$

$y\left(x\right)={\&ExponentialE;}^{x-\mathrm{c\_\_1}}$

$\mathrm{syms}≔\left[\left[1\&comma;0\right]\&comma;\left[0\&comma;\exp \left(x\right)\right]\right]$

$\mathrm{syms}≔\left[\left[1\&comma;0\right]\&comma;\left[0\&comma;{\&ExponentialE;}^x\right]\right]$

$\mathrm{ode}≔\mathrm{equinv}\left(\mathrm{syms}\&comma;y\left(x\right)\&comma;2\right)$

$\mathrm{ode}≔\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=y\left(x\right)+\mathrm{f\_\_1}\left(-y\left(x\right)+\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)$

$\mathrm{reduce\_order}\left(\mathrm{ode}\&comma;\mathrm{syms}\left[1\right]\&comma;u\left(v\right)\right)$

$y\left(x\right)=v\mathbf{where}\left[\left\{\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)=\left(-v-\mathrm{f\_\_1}\left(-v+\frac{1}{u\left(v\right)}\right)\right){u\left(v\right)}^3\right\}\&comma;\left\{v=y\left(x\right)\&comma;u\left(v\right)=\frac{1}{\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)}\right\}\&comma;\left\{x=\int u\left(v\right)\&DifferentialD;v+\mathrm{c\_\_1}\&comma;y\left(x\right)=v\right\}\right]$

$\mathrm{reduce\_order}\left(\mathrm{ode}\&comma;\mathrm{syms}\left[2\right]\&comma;u\left(v\right)\right)$

$y\left(x\right)=\left(\left(\int u\left(v\right)\&DifferentialD;v+\mathrm{c\_\_1}\right){\&ExponentialE;}^v\right)\mathbf{where}\left[\left\{\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)=-2u\left(v\right)+{\&ExponentialE;}^{-v}\mathrm{f\_\_1}\left(u\left(v\right){\&ExponentialE;}^v\right)\right\}\&comma;\left\{v=x\&comma;u\left(v\right)={\&ExponentialE;}^{-x}\left(-y\left(x\right)+\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)\right\}\&comma;\left\{x=v\&comma;y\left(x\right)=\left(\int u\left(v\right)\&DifferentialD;v+\mathrm{c\_\_1}\right){\&ExponentialE;}^v\right\}\right]$

$\mathrm{oo}≔\mathrm{reduce\_order}\left(\mathrm{ode}\&comma;\mathrm{syms}\&comma;u\left(v\right)\&comma;U\left(V\right)\&comma;\mathrm{dif}\&comma;\mathrm{in\_sequence}\right)$

$\mathrm{oo}≔y\left(x\right)=\left(\left(\int u\left(v\right){\&ExponentialE;}^{-v}\&DifferentialD;v+\mathrm{c\_\_1}\right){\&ExponentialE;}^v\right)\mathbf{where}\left[\left\{\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)=-u\left(v\right)+\mathrm{f\_\_1}\left(u\left(v\right)\right)\right\}\&comma;\left\{v=x\&comma;u\left(v\right)=-y\left(x\right)+\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right\}\&comma;\left\{x=v\&comma;y\left(x\right)=\left(\int u\left(v\right){\&ExponentialE;}^{-v}\&DifferentialD;v+\mathrm{c\_\_1}\right){\&ExponentialE;}^v\right\}\right],u\left(v\right)=V\mathbf{where}\left[\left\{U\left(V\right)=-V+\mathrm{f\_\_1}\left(V\right)\right\}\&comma;\left\{V=u\left(v\right)\&comma;U\left(V\right)=\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)\right\}\&comma;\left\{v=\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}\&comma;u\left(v\right)=V\right\}\right]$

$\mathrm{o1}≔\mathrm{op}\left(\left[2\&comma;1\&comma;1\right]\&comma;\mathrm{rhs}\left(\mathrm{oo}\left[1\right]\right)\right)$

$\mathrm{o1}≔\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)=-u\left(v\right)+\mathrm{f\_\_1}\left(u\left(v\right)\right)$

$\mathrm{o2}≔\mathrm{op}\left(\left[2\&comma;1\&comma;1\right]\&comma;\mathrm{rhs}\left(\mathrm{oo}\left[2\right]\right)\right)$

$\mathrm{o2}≔U\left(V\right)=-V+\mathrm{f\_\_1}\left(V\right)$

$\mathrm{o1}≔\mathrm{op}\left(\left[2\&comma;1\&comma;1\right]\&comma;\mathrm{rhs}\left(\mathrm{oo}\left[1\right]\right)\right)$

$\mathrm{o1}≔\frac{\&DifferentialD;}{\&DifferentialD;v}u\left(v\right)=-u\left(v\right)+\mathrm{f\_\_1}\left(u\left(v\right)\right)$

$\mathrm{buildsol}\left(\mathrm{oo}\left[2\right]\&comma;\mathrm{o2}\&comma;\mathrm{implicit}\right)$

$u\left(v\right)=\mathrm{RootOf}\left(-v-\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\mathrm{\_a}-\mathrm{f\_\_1}\left(\mathrm{\_a}\right)}\&DifferentialD;\mathrm{\_a}\right)+\mathrm{c\_\_2}\right)$

$\mathrm{buildsol}\left(\mathrm{oo}\left[1\right]\&comma;\right)$

$y\left(x\right)=\left(\int \mathrm{RootOf}\left(-x-\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\mathrm{\_a}-\mathrm{f\_\_1}\left(\mathrm{\_a}\right)}\&DifferentialD;\mathrm{\_a}\right)+\mathrm{c\_\_2}\right){\&ExponentialE;}^{-x}\&DifferentialD;x+\mathrm{c\_\_1}\right){\&ExponentialE;}^x$

$\mathrm{odetest}\left(\&comma;\mathrm{ode}\right)$

$0$

$\mathrm{buildsol}\left(\mathrm{oo}\right)$

$y\left(x\right)=\left(\left(\int \frac{V{\&ExponentialE;}^{-\left(\int \frac{1}{U\left(V\right)}\&DifferentialD;V\right)-\mathrm{c\_\_2}}}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_1}\right){\&ExponentialE;}^{\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}}\right)\mathbf{where}\left[\left\{U\left(V\right)=-V+\mathrm{f\_\_1}\left(V\right)\right\}\&comma;\left\{V=-y\left(x\right)+\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\&comma;U\left(V\right)=-\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)+\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right\}\&comma;\left\{x=\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}\&comma;y\left(x\right)=\left(\int \frac{V{\&ExponentialE;}^{-\left(\int \frac{1}{U\left(V\right)}\&DifferentialD;V\right)-\mathrm{c\_\_2}}}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_1}\right){\&ExponentialE;}^{\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}}\right\}\right]$

$\mathrm{op}\left(\left[2\&comma;1\&comma;1\right]\&comma;\mathrm{rhs}\left(\right)\right)$

$U\left(V\right)=-V+\mathrm{f\_\_1}\left(V\right)$

$\mathrm{op}\left(\left[2\&comma;3\right]\&comma;\mathrm{rhs}\left(\right)\right)$

$\left\{x=\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}\&comma;y\left(x\right)=\left(\int \frac{V{\&ExponentialE;}^{-\left(\int \frac{1}{U\left(V\right)}\&DifferentialD;V\right)-\mathrm{c\_\_2}}}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_1}\right){\&ExponentialE;}^{\int \frac{1}{U\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}}\right\}$

$\mathrm{subs}\left(\&comma;\right)$

$\left\{x=\int \frac{1}{-V+\mathrm{f\_\_1}\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}\&comma;y\left(x\right)=\left(\int \frac{V{\&ExponentialE;}^{-\left(\int \frac{1}{-V+\mathrm{f\_\_1}\left(V\right)}\&DifferentialD;V\right)-\mathrm{c\_\_2}}}{-V+\mathrm{f\_\_1}\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_1}\right){\&ExponentialE;}^{\int \frac{1}{-V+\mathrm{f\_\_1}\left(V\right)}\&DifferentialD;V+\mathrm{c\_\_2}}\right\}$