Lienard - Maple Help

Lienard ODEs

Description

 • The general form of the Lienard ODE is given by the following:
 > Lienard_ode := diff(y(x),x,x)+f(x)*diff(y(x),x)+y(x)=0;
 ${\mathrm{Lienard_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{f}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}$ (1)
 where f(x) is an arbitrary function of x. See Villari, "Periodic Solutions of Lienard's Equation".
 • All linear second order homogeneous ODEs can be transformed into first order ODEs of Riccati type. That can be done by giving the symmetry [0,y] to dsolve (all linear homogeneous ODEs have this symmetry) or just calling convert (see convert,ODEs).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right):$
 > $\mathrm{odeadvisor}\left(\mathrm{Lienard_ode}\right)$
 $\left[{\mathrm{_Lienard}}\right]$ (2)

Reduction to Riccati by giving the symmetry to dsolve

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Lienard_ode},\mathrm{HINT}=\left[0,y\right]\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}\left({{ⅇ}}^{{\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right){=}{-}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}{-}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){}{f}{}\left({\mathrm{_a}}\right){-}{1}\right\}{,}\left\{{\mathrm{_a}}{=}{x}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)}{{y}{}\left({x}\right)}\right\}{,}\left\{{x}{=}{\mathrm{_a}}{,}{y}{}\left({x}\right){=}{{ⅇ}}^{{\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right\}\right]$ (3)

The reduced ODE above is of Riccati type

 > $\mathrm{reduced_ode}≔\mathrm{op}\left(\left[2,2,1,1\right],\mathrm{ans}\right)$
 ${\mathrm{reduced_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right){=}{-}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}{-}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){}{f}{}\left({\mathrm{_a}}\right){-}{1}$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{reduced_ode}\right)$
 $\left[{\mathrm{_Riccati}}\right]$ (5)

Converting this ODE into a first order ODE of Riccati type

 > $\mathrm{Riccati_ode_TR}≔\mathrm{convert}\left(\mathrm{Lienard_ode},\mathrm{Riccati}\right)$
 ${\mathrm{Riccati_ode_TR}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_a}{}\left({x}\right){=}{\mathrm{_F1}}{}\left({x}\right){}{{\mathrm{_a}}{}\left({x}\right)}^{{2}}{+}\frac{\left({-}{f}{}\left({x}\right){}{\mathrm{_F1}}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_F1}}{}\left({x}\right)\right){}{\mathrm{_a}}{}\left({x}\right)}{{\mathrm{_F1}}{}\left({x}\right)}{+}\frac{{1}}{{\mathrm{_F1}}{}\left({x}\right)}{,}\left\{{y}{}\left({x}\right){=}{{ⅇ}}^{{-}\left({\int }{\mathrm{_a}}{}\left({x}\right){}{\mathrm{_F1}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{}{\mathrm{_C1}}\right\}$ (6)

In the answer returned by convert, there are the Riccati ODE and the transformation of the variable used. Changes of variables in ODEs can be performed using ?PDEtools[dchange]. For example, using the transformation of variables above, we can recover the result returned by convert.

 > $\mathrm{TR}≔\mathrm{Riccati_ode_TR}\left[2\right]$
 ${\mathrm{TR}}{≔}\left\{{y}{}\left({x}\right){=}{{ⅇ}}^{{-}\left({\int }{\mathrm{_a}}{}\left({x}\right){}{\mathrm{_F1}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{}{\mathrm{_C1}}\right\}$ (7)
 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{dchange}\right)$
 $\left[{\mathrm{dchange}}\right]$ (8)
 > $\mathrm{collect}\left(\mathrm{isolate}\left(\mathrm{dchange}\left(\mathrm{TR},\mathrm{Lienard_ode},\left[\mathrm{_a}\left(x\right)\right]\right),\mathrm{diff}\left(\mathrm{_a}\left(x\right),x\right)\right),\mathrm{_a}\left(x\right),\mathrm{normal}\right)$
 $\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_a}{}\left({x}\right){=}{\mathrm{_F1}}{}\left({x}\right){}{{\mathrm{_a}}{}\left({x}\right)}^{{2}}{-}\frac{\left({f}{}\left({x}\right){}{\mathrm{_F1}}{}\left({x}\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_F1}}{}\left({x}\right)\right){}{\mathrm{_a}}{}\left({x}\right)}{{\mathrm{_F1}}{}\left({x}\right)}{+}\frac{{1}}{{\mathrm{_F1}}{}\left({x}\right)}$ (9)