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](..).

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

$\mathrm{ode}\left[1\right]≔x^{n-1}{\mathrm{diff}\left(y\left(x\right)\,x\right)}^n-nx\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)$

${\mathrm{ode}}_1≔x^{n-1}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^n-nx\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)$

$\mathrm{parametricsol}\left(\mathrm{ode}\left[1\right]\right)$

$\left\{\left[y\left(\mathrm{\_T}\right)=-{\left(\mathrm{c\_\_1}{\mathrm{\_T}}^{-\frac{n}{n-1}}\right)}^{n-1}{\mathrm{\_T}}^n+n\mathrm{c\_\_1}{\mathrm{\_T}}^{-\frac{1}{n-1}}\,x\left(\mathrm{\_T}\right)=\mathrm{c\_\_1}{\mathrm{\_T}}^{-\frac{n}{n-1}}\right]\right\}$

$\mathrm{ans}\left[1\right]≔\mathrm{parametricsol}\left(\mathrm{ode}\left[1\right]\,\mathrm{explicit}\right)$

${\mathrm{ans}}_1≔\left\{y\left(x\right)=-x^{n-1}{\left(\frac{\mathrm{c\_\_1}{\left(\frac{x}{\mathrm{c\_\_1}}\right)}^{\frac{1}{n}}}{x}\right)}^n+n\mathrm{c\_\_1}{\left(\frac{x}{\mathrm{c\_\_1}}\right)}^{\frac{1}{n}}\right\}$

$\mathrm{odetest}\left(\mathrm{ans}\left[1\right]\,\mathrm{ode}\left[1\right]\right)$

$0$

$\mathrm{ode}\left[2\right]≔\mathrm{diff}\left(y\left(x\right)\,x\right)=F\left(\frac{y\left(x\right)-x\ln \left(x\right)}{x}\right)+\ln \left(x\right)$

${\mathrm{ode}}_2≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=F\left(\frac{y\left(x\right)-x\ln \left(x\right)}{x}\right)+\ln \left(x\right)$

$\mathrm{parametricsol}\left(\mathrm{ode}\left[2\right]\,\mathrm{Lie}\right)$

$\left\{\left[y\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{1}{F\left(\mathrm{\_T}\right)-\mathrm{\_T}-1}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}}\left(\int \frac{1}{F\left(\mathrm{\_T}\right)-\mathrm{\_T}-1}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}+\mathrm{\_T}\right)\,x\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{1}{F\left(\mathrm{\_T}\right)-\mathrm{\_T}-1}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}}\right]\right\}$

$\mathrm{ode}\left[3\right]≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)={y\left(x\right)}^2\mathrm{diff}\left(y\left(x\right)\,x\right)$

${\mathrm{ode}}_3≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)={y\left(x\right)}^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{parametricsol}\left(\mathrm{ode}\left[3\right]\right)$

$\left\{\left[y\left(\mathrm{\_T}\right)=\frac{{\mathrm{\_T}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{c\_\_2}}{{\left(\mathrm{\_T}-3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\,x\left(\mathrm{\_T}\right)=\int -\frac{1}{{\mathrm{\_T}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(\mathrm{\_T}-3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\mathrm{c\_\_2}}^2}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}\right]\right\}$