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 parametricsol
 computes parametric solutions for Ordinary Differential Equations

 Calling Sequence parametricsol(ODE, v, Lie, explicit, )

Parameters

 ODE - first order differential equation v - optiona, dependent variable of the ODE, required when not obvious Lie - optional, to request the use of symmetry methods when computing parametric solutions for 1st order ODEs explicit - optional, for 1st order ODEs, to request the removal of the parameter used to express the parametric solution - the optional arguments explained in dsolve,Lie, in connection with using Lie methods to compute the parametric solution

Description

 • parametricsol computes parametric solutions mainly for nonlinear ODEs, although the methods implemented applies as well to linear ODEs. Two methods are implemented:
 – Some 1st order ODEs fit a pattern for which a parametric solution, as explained in odeadvisor[parametric], can be sought. This is the default method tried for 1st order ODEs.
 – More general, 1st and higher order ODEs, for which a as many point symmetries as the differential order, forming a group, can be computed, can always have their general solution represented in parametric form (see DEtools[reduce_order]). This is the default method for 2nd and higher order ODEs, and is invoked for 1st order ODEs using the optional argument Lie
 • This function is part of the DEtools package, and so it can be used in the form parametricsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[parametricsol](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > ${\mathrm{ode}}_{1}≔{x}^{n-1}{\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)}^{n}-nx\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+y\left(x\right)$
 ${{\mathrm{ode}}}_{{1}}{≔}{{x}}^{{n}{-}{1}}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{n}}{-}{n}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right)$ (1)
 > $\mathrm{parametricsol}\left({\mathrm{ode}}_{1}\right)$
 $\left\{\left[{y}{}\left({\mathrm{_T}}\right){=}{-}\frac{{\left({\mathrm{_C1}}{}{{\mathrm{_T}}}^{{-}\frac{{n}}{{n}{-}{1}}}\right)}^{{n}}{}{{\mathrm{_T}}}^{\frac{{{n}}^{{2}}}{{n}{-}{1}}}}{{\mathrm{_C1}}}{+}{n}{}{\mathrm{_C1}}{}{{\mathrm{_T}}}^{{-}\frac{{1}}{{n}{-}{1}}}{,}{x}{}\left({\mathrm{_T}}\right){=}{\mathrm{_C1}}{}{{\mathrm{_T}}}^{{-}\frac{{n}}{{n}{-}{1}}}\right]\right\}$ (2)

To see the solution to this equation without the parameter _T use the optional argument explicit

 > $\mathrm{parametricsol}\left({\mathrm{ode}}_{1},\mathrm{explicit}\right)$
 $\left\{{y}{}\left({x}\right){=}{\mathrm{_C1}}{}{n}{}{\left(\frac{{x}}{{\mathrm{_C1}}}\right)}^{\frac{{1}}{{n}}}{-}\frac{{\left(\frac{{1}}{{\mathrm{_C1}}}\right)}^{{-}{n}}}{{\mathrm{_C1}}}\right\}$ (3)

A 1st order ODE example where the solution can only be computed using symmetry methods

 > ${\mathrm{ode}}_{2}≔\frac{ⅆ}{ⅆx}y\left(x\right)=F\left(\frac{y\left(x\right)-x\mathrm{ln}\left(x\right)}{x}\right)+\mathrm{ln}\left(x\right)$
 ${{\mathrm{ode}}}_{{2}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left(\frac{{y}{}\left({x}\right){-}{x}{}{\mathrm{ln}}{}\left({x}\right)}{{x}}\right){+}{\mathrm{ln}}{}\left({x}\right)$ (4)
 > $\mathrm{parametricsol}\left({\mathrm{ode}}_{2},\mathrm{Lie}\right)$
 $\left\{\left[{y}{}\left({\mathrm{_T}}\right){=}{{ⅇ}}^{{\int }\frac{{1}}{{F}{}\left({\mathrm{_T}}\right){-}{\mathrm{_T}}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}}{}\left({\int }\frac{{1}}{{F}{}\left({\mathrm{_T}}\right){-}{\mathrm{_T}}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}{+}{\mathrm{_T}}\right){,}{x}{}\left({\mathrm{_T}}\right){=}{{ⅇ}}^{{\int }\frac{{1}}{{F}{}\left({\mathrm{_T}}\right){-}{\mathrm{_T}}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}}\right]\right\}$ (5)

A 2nd order nonlinear ODE; Lie methods are automatically invoked when the differential order is higher than 1

 > ${\mathrm{ode}}_{3}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)={y\left(x\right)}^{2}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)$
 ${{\mathrm{ode}}}_{{3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{y}{}\left({x}\right)}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (6)
 > $\mathrm{parametricsol}\left({\mathrm{ode}}_{3}\right)$
 $\left\{\left[{y}{}\left({\mathrm{_T}}\right){=}\frac{{{\mathrm{_T}}}^{{1}}{{3}}}{}{\mathrm{_C2}}}{{\left({\mathrm{_T}}{-}{3}\right)}^{{1}}{{3}}}}{,}{x}{}\left({\mathrm{_T}}\right){=}{\int }{-}\frac{{1}}{{\left({\mathrm{_T}}{-}{3}\right)}^{{1}}{{3}}}{}{{\mathrm{_T}}}^{{2}}{{3}}}{}{{\mathrm{_C2}}}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}\right]\right\}$ (7)