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;

$\mathrm{Lienard\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)=0$

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).

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

$\mathrm{odeadvisor}\left(\mathrm{Lienard\_ode}\right)$

$\left[\mathrm{\_Lienard}\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Lienard\_ode}\,\mathrm{HINT}=\left[0\,y\right]\right)$

$\mathrm{ans}≔y\left(x\right)=\left({\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right)\mathbf{where}\left[\left\{\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=-{\mathrm{\_b}\left(\mathrm{\_a}\right)}^2-\mathrm{\_b}\left(\mathrm{\_a}\right)f\left(\mathrm{\_a}\right)-1\right\}\,\left\{\mathrm{\_a}=x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{y\left(x\right)}\right\}\,\left\{x=\mathrm{\_a}\,y\left(x\right)={\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right\}\right]$

$\mathrm{reduced\_ode}≔\mathrm{op}\left(\left[2\,2\,1\,1\right]\,\mathrm{ans}\right)$

$\mathrm{reduced\_ode}≔\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=-{\mathrm{\_b}\left(\mathrm{\_a}\right)}^2-\mathrm{\_b}\left(\mathrm{\_a}\right)f\left(\mathrm{\_a}\right)-1$

$\mathrm{odeadvisor}\left(\mathrm{reduced\_ode}\right)$

$\left[\mathrm{\_Riccati}\right]$

$\mathrm{Riccati\_ode\_TR}≔\mathrm{convert}\left(\mathrm{Lienard\_ode}\,\mathrm{Riccati}\right)$

$\mathrm{Riccati\_ode\_TR}≔\frac{\ⅆ}{\ⅆx}\mathrm{\_a}\left(x\right)=\mathrm{\_F1}\left(x\right){\mathrm{\_a}\left(x\right)}^2+\frac{\left(-f\left(x\right)\mathrm{\_F1}\left(x\right)-\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\mathrm{\_a}\left(x\right)}{\mathrm{\_F1}\left(x\right)}+\frac{1}{\mathrm{\_F1}\left(x\right)},\left\{y\left(x\right)={\ⅇ}^{-\left(\int \mathrm{\_a}\left(x\right)\mathrm{\_F1}\left(x\right)\ⅆx\right)}\mathrm{c\_\_1}\right\}$

$\mathrm{TR}≔\mathrm{Riccati\_ode\_TR}\left[2\right]$

$\mathrm{TR}≔\left\{y\left(x\right)={\ⅇ}^{-\left(\int \mathrm{\_a}\left(x\right)\mathrm{\_F1}\left(x\right)\ⅆx\right)}\mathrm{c\_\_1}\right\}$

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{dchange}\right)$

$\left[\mathrm{dchange}\right]$

$\mathrm{collect}\left(\mathrm{isolate}\left(\mathrm{dchange}\left(\mathrm{TR}\,\mathrm{Lienard\_ode}\,\left[\mathrm{\_a}\left(x\right)\right]\right)\,\mathrm{diff}\left(\mathrm{\_a}\left(x\right)\,x\right)\right)\,\mathrm{\_a}\left(x\right)\,\mathrm{normal}\right)$

$\frac{\ⅆ}{\ⅆx}\mathrm{\_a}\left(x\right)=\mathrm{\_F1}\left(x\right){\mathrm{\_a}\left(x\right)}^2-\frac{\left(f\left(x\right)\mathrm{\_F1}\left(x\right)+\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\mathrm{\_a}\left(x\right)}{\mathrm{\_F1}\left(x\right)}+\frac{1}{\mathrm{\_F1}\left(x\right)}$