odeadvisor/Liouville

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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 := diff(y(x), `$`(x, 2))+g(y(x))*(diff(y(x), x))^2+f(x)*(diff(y(x), x)) = 0

(1.1)

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

Examples

(2.1)

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

symmetries := [_xi = 0, _eta = exp(-(Int(g(y), y)))], [_xi = 0, _eta = (Int(exp(Int(g(y), y)), y))*exp(-(Int(g(y), y)))], [_xi = exp(-(Int(-f(x), x))), _eta = 0], [_xi = (Int(exp(Int(-f(x), x)), x))*exp(-(Int(-f(x), x))), _eta = 0]

(2.2)

These symmetries can be tested using symtest

(2.3)

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

ans := Int(exp(Int(g(_b), _b)), _b = `` .. y(x))-_C1*(Int(exp(-(Int(f(x), x))), x))-_C2 = 0

(2.4)

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

(2.5)

	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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