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Lienard ODEs
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Description
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The general form of the Lienard ODE is given by the following:
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Lienard_ode := diff(y(x),x,x)+f(x)*diff(y(x),x)+y(x)=0;
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where f(x) is an arbitrary function of x. See Villari, "Periodic Solutions of Lienard's Equation".
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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).
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Examples
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![[_Lienard]](/support/helpjp/helpview.aspx?si=7243/file04358/math43.png)
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Reduction to Riccati by giving the symmetry to dsolve
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![ans := dsolve(Lienard_ode, HINT = [0, y])](/support/helpjp/helpview.aspx?si=7243/file04358/math52.png)
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![ans := y(x) = `&where`(exp(Int(_b(_a), _a)+_C1), [{diff(_b(_a), _a) = -_b(_a)^2-f(_a)*_b(_a)-1}, {_a = x, _b(_a) = (diff(y(x), x))/y(x)}, {x = _a, y(x) = exp(Int(_b(_a), _a)+_C1)}])](/support/helpjp/helpview.aspx?si=7243/file04358/math55.png)
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The reduced ODE above is of Riccati type
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![reduced_ode := op([2, 2, 1, 1], ans)](/support/helpjp/helpview.aspx?si=7243/file04358/math61.png)
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![[_Riccati]](/support/helpjp/helpview.aspx?si=7243/file04358/math71.png)
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Converting this ODE into a first order ODE of Riccati type
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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.
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![TR := Riccati_ode_TR[2]](/support/helpjp/helpview.aspx?si=7243/file04358/math92.png)
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![[dchange]](/support/helpjp/helpview.aspx?si=7243/file04358/math102.png)
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![collect(isolate(dchange(TR, Lienard_ode, [_a(x)]), diff(_a(x), x)), _a(x), normal)](/support/helpjp/helpview.aspx?si=7243/file04358/math106.png)
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See Also
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DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.
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