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Van der Pol ODEs
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Description
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The general form of the Van der Pol ODE is given by the following:
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Van_der_Pol_ode := diff(y(x),x,x)-mu*(1-y(x)^2)*diff(y(x),x)+y(x)=0;
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| (1) |
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See Birkhoff and Rota, "Ordinary Differential Equations", p. 134.
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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):
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with(DEtools, odeadvisor, symgen):
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odeadvisor(Van_der_Pol_ode);
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![[[_2nd_order, _missing_x], _Van_der_Pol]](/support/helpjp/helpview.aspx?si=7244/file04363/math33.png)
| (2) |
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symgen(Van_der_Pol_ode, way=3);
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![[_xi = 1, _eta = 0]](/support/helpjp/helpview.aspx?si=7244/file04363/math40.png)
| (3) |
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From which, giving the same indication directly to dsolve you obtain the reduction of order
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ans := dsolve(Van_der_Pol_ode,way=3);
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![ans := y(x) = `&where`(_a, [{(diff(_b(_a), _a))*_b(_a)-mu*_b(_a)+mu*_b(_a)*_a^2+_a = 0}, {_a = y(x), _b(_a) = diff(y(x), x)}, {x = Int(1/_b(_a), _a)+_C1, y(x) = _a}])](/support/helpjp/helpview.aspx?si=7244/file04363/math52.png)
| (4) |
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For the structure of the solution above see ?ODESolStruc. Reductions of order can also be tested with odetest
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odetest(ans,Van_der_Pol_ode);
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| (5) |
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The reduced ODE is of type Abel, and can be selected using either the mouse, or the following:
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reduced_ode := op([2,2,1,1],ans);
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| (6) |
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odeadvisor(reduced_ode);
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![[_rational, [_Abel, `2nd type`, `class A`]]](/support/helpjp/helpview.aspx?si=7244/file04363/math83.png)
| (7) |
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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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