Van der Pol ODEs - Maple Help

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Van der Pol ODEs

Description

• 

The general form of the Van der Pol ODE is given by the following:

Van_der_Pol_ode := diff(y(x),x,x)-mu*(1-y(x)^2)*diff(y(x),x)+y(x)=0;

Van_der_Pol_ode:=ⅆ2ⅆx2yxμ1yx2ⅆⅆxyx+yx=0

(1)
  

See Birkhoff and Rota, "Ordinary Differential Equations", p. 134.

  

The second order Van der Pol ODE can be reduced to a first order ODE of Abel type as soon as the system succeeds in finding one polynomial symmetry for it (see symgen):

with(DEtools, odeadvisor, symgen):

odeadvisor(Van_der_Pol_ode);

_2nd_order,_missing_x,_Van_der_Pol

(2)

symgen(Van_der_Pol_ode, way=3);

_ξ=1,_η=0

(3)
  

From which, giving the same indication directly to dsolve you obtain the reduction of order

ans := dsolve(Van_der_Pol_ode,way=3);

ans:=yx=_a &where ⅆⅆ_a_b_a_b_a+_b_a_a2μμ_b_a+_a=0,_a=yx,_b_a=ⅆⅆxyx,x=∫1_b_aⅆ_a+_C1,yx=_a

(4)
  

For the structure of the solution above see ODESolStruc. Reductions of order can also be tested with odetest

odetest(ans,Van_der_Pol_ode);

0

(5)
  

The reduced ODE is of type Abel, and can be selected using either the mouse, or the following:

reduced_ode := op([2,2,1,1],ans);

reduced_ode:=ⅆⅆ_a_b_a_b_a+_b_a_a2μμ_b_a+_a=0

(6)

odeadvisor(reduced_ode);

_rational,_Abel,2nd type,class A

(7)

See Also

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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