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Emden, Modified Emden, and Emden/Fowler ODEs
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Description
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The general forms of the Emden, modified Emden and Emden/Fowler ODEs are given by the following:
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Emden_ode := diff(x^2*diff(y(x),x),x)+x^2*y(x)^n=0;
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| (1) |
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modified_Emden_ode := diff(diff(y(x),x),x)+a(x)*diff(y(x),x)+y(x)^n = 0;
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| (2) |
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Emden_Fowler_ode := diff(x^p*diff(y(x),x),x)+x^sigma*y(x)^n=0;
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| (3) |
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where n is an integer and a(x) is an arbitrary function of x.
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See Leach, "First Integrals for the modified Emden equation"; and Rosenau, "A Note on Integration of the Emden-Fowler Equation". There are certain special cases of the Emden-Fowler equation which can be solved exactly. See also Polyanin and Zaitsev, "Exact Solutions of Ordinary Differential Equations", p. 241.
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Examples
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![[_Emden, [_2nd_order, _with_linear_symmetries]]](/support/helpjp/helpview.aspx?si=7237/file04351/math48.png)
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![[[_Emden, _modified]]](/support/helpjp/helpview.aspx?si=7237/file04351/math55.png)
| (5) |
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![[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]](/support/helpjp/helpview.aspx?si=7237/file04351/math62.png)
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The second order Emden ODE can be reduced to a first order Abel ODE once the system succeeds in finding one polynomial symmetry for it (see ?symgen):
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![[_xi = -(1/2)*x*(n-1), _eta = y]](/support/helpjp/helpview.aspx?si=7237/file04351/math74.png)
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From which, giving the same indication directly to dsolve (see ?dsolve,Lie) it returns a reduced (Abel type) ODE:
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![ans := dsolve(Emden_ode, HINT = [-(1/2)*x*n+(1/2)*x, y])](/support/helpjp/helpview.aspx?si=7237/file04351/math86.png)
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![ans := y(x) = `&where`(_a*exp(Int(_b(_a), _a)+_C1), [{diff(_b(_a), _a) = ((1/4)*_a^n*n^2+(1/4)*_a^n+(3/2)*_a-(1/2)*n*_a-(1/2)*_a^n*n)*_b(_a)^3+(-(1/2)*n+5/2)*_b(_a)^2}, {_a = y(x)*x^(2/(n-1)), _b(_a) = -2/(x^(2/(n-1))*((diff(y(x), x))*x*n-x*(diff(y(x), x))+2*y(x)))}, {x = exp(-(1/2)*(Int(_b(_a), _a)+_C1)*n+(1/2)*(Int(_b(_a), _a))+(1/2)*_C1), y(x) = _a*exp(Int(_b(_a), _a)+_C1)}])](/support/helpjp/helpview.aspx?si=7237/file04351/math89.png)
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The reduced ODE can be selected using the mouse, or through:
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![reduced_ode := op([2, 2, 1, 1], ans)](/support/helpjp/helpview.aspx?si=7237/file04351/math95.png)
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| (9) |
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![[_Abel]](/support/helpjp/helpview.aspx?si=7237/file04351/math105.png)
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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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