Van der Pol ODEs - Maple Help

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Van der Pol ODEs



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;



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):




symgen(Van_der_Pol_ode, way=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


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




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);




_rational,_Abel,2nd type,class A


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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