Liouville ODEs - Maple Help

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Liouville ODEs

Description

• 

The general form of the Liouville ODE is given by the following:

Liouville_ode := diff(y(x),x,x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0;

Liouville_ode:=ⅆ2ⅆx2yx+gyxⅆⅆxyx2+fxⅆⅆxyx=0

(1)
  

where g and f are arbitrary functions. See Goldstein and Braun, "Advanced Methods for the Solution of Differential Equations".

Examples

withDEtools,odeadvisor,symgen,symtest:

odeadvisorLiouville_ode

_Liouville,_2nd_order,_reducible,_mu_x_y1,_2nd_order,_reducible,_mu_xy

(2)

The Liouville ODE has the following symmetries (see dsolve,Lie):

symmetries:=symgenLiouville_ode

symmetries:=_ξ=0,_η=ⅇ∫gyⅆy,_ξ=0,_η=∫ⅇ∫gyⅆyⅆyⅇ∫gyⅆy,_ξ=ⅇ∫fxⅆx,_η=0,_ξ=∫ⅇ∫fxⅆxⅆxⅇ∫fxⅆx,_η=0

(3)

These symmetries can be tested using symtest

mapsymtest,symmetries,Liouville_ode

0,0,0,0

(4)

Knowing two independent symmetries for a second order ODE almost always leads to its answer, as in the following Liouville ODE:

ans:=dsolveLiouville_ode

ans:=∫yxⅇ∫g_bⅆ_bⅆ_b_C1∫ⅇ∫fxⅆxⅆx_C2=0

(5)

Implicit and explicit answers for ODEs can be tested using odetest.

odetestans,Liouville_ode

0

(6)

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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