Emden, Modified Emden, and Emden/Fowler ODEs - Maple Programming Help

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Emden, Modified Emden, and Emden/Fowler ODEs

 

Description

Examples

Description

• 

The general forms of the Emden, modified Emden and Emden/Fowler ODEs are given by the following:

Emden_ode := diff(x^2*diff(y(x),x),x)+x^2*y(x)^n=0;

Emden_ode:=2xⅆⅆxyx+x2ⅆ2ⅆx2yx+x2yxn=0

(1)

modified_Emden_ode := diff(diff(y(x),x),x)+a(x)*diff(y(x),x)+y(x)^n = 0;

modified_Emden_ode:=ⅆ2ⅆx2yx+axⅆⅆxyx+yxn=0

(2)

Emden_Fowler_ode := diff(x^p*diff(y(x),x),x)+x^sigma*y(x)^n=0;

Emden_Fowler_ode:=xppⅆⅆxyxx+xpⅆ2ⅆx2yx+xσyxn=0

(3)
  

where n is an integer and a(x) is an arbitrary function of x.

  

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.

Examples

withDEtools,odeadvisor,symgen:

odeadvisorEmden_ode

_Emden,_2nd_order,_with_linear_symmetries

(4)

odeadvisormodified_Emden_ode

_Emden,_modified

(5)

odeadvisorEmden_Fowler_ode

_Emden,_Fowler,_2nd_order,_with_linear_symmetries

(6)

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):

symgenEmden_ode,way=3

_ξ=12xn1,_η=y

(7)

From which, giving the same indication directly to dsolve (see dsolve/Lie) it returns a reduced (Abel type) ODE:

ansdsolveEmden_ode,HINT=1xn2+1x2,y

ans:=yx=_aⅇ∫_b_aⅆ_a+_C1 &where ⅆⅆ_a_b_a=14_ann212_ann12_an+14_an+32_a_b_a3+12n+52_b_a2,_a=yxx2n1,_b_a=2x2n1ⅆⅆxyxxnxⅆⅆxyx+2yx,x=ⅇ12∫_b_aⅆ_a+_C1n+12∫_b_aⅆ_a+12_C1,yx=_aⅇ∫_b_aⅆ_a+_C1

(8)

The reduced ODE can be selected using the mouse, or through:

reduced_odeop2,2,1,1,ans

reduced_ode:=ⅆⅆ_a_b_a=14_ann212_ann12_an+14_an+32_a_b_a3+12n+52_b_a2

(9)

odeadvisorreduced_ode

_Abel

(10)

See Also

DEtools

odeadvisor

dsolve

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

odeadvisor,types

 


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