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

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Emden, Modified Emden, and Emden/Fowler ODEs

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;
 ${\mathrm{Emden_ode}}{:=}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}{{x}}^{{2}}{}{{y}{}\left({x}\right)}^{{n}}{=}{0}$ (1)
 > modified_Emden_ode := diff(diff(y(x),x),x)+a(x)*diff(y(x),x)+y(x)^n = 0;
 ${\mathrm{modified_Emden_ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}{a}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{{y}{}\left({x}\right)}^{{n}}{=}{0}$ (2)
 > Emden_Fowler_ode := diff(x^p*diff(y(x),x),x)+x^sigma*y(x)^n=0;
 ${\mathrm{Emden_Fowler_ode}}{:=}\frac{{{x}}^{{p}}{}{p}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}{{x}}{+}{{x}}^{{p}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}{{x}}^{{\mathrm{σ}}}{}{{y}{}\left({x}\right)}^{{n}}{=}{0}$ (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

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen}\right):$
 > $\mathrm{odeadvisor}\left(\mathrm{Emden_ode}\right)$
 $\left[{\mathrm{_Emden}}{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{modified_Emden_ode}\right)$
 $\left[\left[{\mathrm{_Emden}}{,}{\mathrm{_modified}}\right]\right]$ (5)
 > $\mathrm{odeadvisor}\left(\mathrm{Emden_Fowler_ode}\right)$
 $\left[\left[{\mathrm{_Emden}}{,}{\mathrm{_Fowler}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (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):

 > $\mathrm{symgen}\left(\mathrm{Emden_ode},\mathrm{way}=3\right)$
 $\left[{\mathrm{_ξ}}{=}{-}\frac{{1}}{{2}}{}{x}{}\left({n}{-}{1}\right){,}{\mathrm{_η}}{=}{y}\right]$ (7)

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

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Emden_ode},\mathrm{HINT}=\left[-\frac{1xn}{2}+\frac{1x}{2},y\right]\right)$
 ${\mathrm{ans}}{:=}{y}{}\left({x}\right){=}\left({\mathrm{_a}}{}{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right){&where}\left[\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\left(\frac{{1}}{{4}}{}{{\mathrm{_a}}}^{{n}}{}{{n}}^{{2}}{-}\frac{{1}}{{2}}{}{{\mathrm{_a}}}^{{n}}{}{n}{-}\frac{{1}}{{2}}{}{\mathrm{_a}}{}{n}{+}\frac{{1}}{{4}}{}{{\mathrm{_a}}}^{{n}}{+}\frac{{3}}{{2}}{}{\mathrm{_a}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}{+}\left({-}\frac{{1}}{{2}}{}{n}{+}\frac{{5}}{{2}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}\right\}{,}\left\{{\mathrm{_a}}{=}{y}{}\left({x}\right){}{{x}}^{\frac{{2}}{{n}{-}{1}}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}{-}\frac{{2}}{{{x}}^{\frac{{2}}{{n}{-}{1}}}{}\left(\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){}{x}{}{n}{-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right)\right)}\right\}{,}\left\{{x}{=}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\left({∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}\right){}{n}{+}\frac{{1}}{{2}}{}{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}\frac{{1}}{{2}}{}{\mathrm{_C1}}}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}{}{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right\}\right]$ (8)

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

 > $\mathrm{reduced_ode}≔\mathrm{op}\left(\left[2,2,1,1\right],\mathrm{ans}\right)$
 ${\mathrm{reduced_ode}}{:=}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\left(\frac{{1}}{{4}}{}{{\mathrm{_a}}}^{{n}}{}{{n}}^{{2}}{-}\frac{{1}}{{2}}{}{{\mathrm{_a}}}^{{n}}{}{n}{-}\frac{{1}}{{2}}{}{\mathrm{_a}}{}{n}{+}\frac{{1}}{{4}}{}{{\mathrm{_a}}}^{{n}}{+}\frac{{3}}{{2}}{}{\mathrm{_a}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}{+}\left({-}\frac{{1}}{{2}}{}{n}{+}\frac{{5}}{{2}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}$ (9)
 > $\mathrm{odeadvisor}\left(\mathrm{reduced_ode}\right)$
 $\left[{\mathrm{_Abel}}\right]$ (10)