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Liouville ODEs
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Description
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The general form of the Liouville ODE is given by the following:
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Liouville_ode := diff(y(x),x,x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0;
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where g and f are arbitrary functions. See Goldstein and Braun, "Advanced Methods for the Solution of Differential Equations".
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Examples
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![[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]](/support/helpjp/helpview.aspx?si=7242/file04359/math32.png)
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The Liouville ODE has the following symmetries (see dsolve,Lie):
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![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]](/support/helpjp/helpview.aspx?si=7242/file04359/math44.png)
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These symmetries can be tested using symtest
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![map(symtest, [symmetries], Liouville_ode)](/support/helpjp/helpview.aspx?si=7242/file04359/math53.png)
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![[0, 0, 0, 0]](/support/helpjp/helpview.aspx?si=7242/file04359/math56.png)
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Knowing two independent symmetries for a second order ODE almost always leads to its answer, as in the following Liouville ODE:
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Implicit and explicit answers for ODEs can be tested using odetest.
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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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