Lienard ODEs - Maple Programming Help

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Lienard ODEs

 

Description

Examples

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;

Lienard_ode:=ⅆ2ⅆx2yx+fxⅆⅆxyx+yx=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

withDEtools,odeadvisor:

odeadvisorLienard_ode

_Lienard

(2)

Reduction to Riccati by giving the symmetry to dsolve

ansdsolveLienard_ode,HINT=0,y

ans:=yx=ⅇ∫_b_aⅆ_a+_C1 &where ⅆⅆ_a_b_a=_b_a2_b_af_a1,_a=x,_b_a=ⅆⅆxyxyx,x=_a,yx=ⅇ∫_b_aⅆ_a+_C1

(3)

The reduced ODE above is of Riccati type

reduced_odeop2,2,1,1,ans

reduced_ode:=ⅆⅆ_a_b_a=_b_a2_b_af_a1

(4)

odeadvisorreduced_ode

_Riccati

(5)

Converting this ODE into a first order ODE of Riccati type

Riccati_ode_TRconvertLienard_ode,Riccati

Riccati_ode_TR:=ⅆⅆx_ax=_F1x_ax2+fx_F1xⅆⅆx_F1x_ax_F1x+1_F1x,yx=ⅇ∫_ax_F1xⅆx_C1

(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.

TRRiccati_ode_TR2

TR:=yx=ⅇ∫_ax_F1xⅆx_C1

(7)

withPDEtools,dchange

dchange

(8)

collectisolatedchangeTR,Lienard_ode,_ax,ⅆⅆx_ax,_ax,normal

ⅆⅆx_ax=_F1x_ax2fx_F1x+ⅆⅆx_F1x_ax_F1x+1_F1x

(9)

See Also

DEtools

odeadvisor

dsolve

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

odeadvisor,types

 


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