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;

$\mathrm{Emden\_ode}≔2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x^2{y\left(x\right)}^n=0$

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

$\mathrm{modified\_Emden\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+a\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{y\left(x\right)}^n=0$

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

$\mathrm{Emden\_Fowler\_ode}≔\frac{x^{p}p\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{x}+x^p\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x^{\mathrm{\sigma }}{y\left(x\right)}^n=0$

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.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\,\mathrm{symgen}\right)\:$

$\mathrm{odeadvisor}\left(\mathrm{Emden\_ode}\right)$

$\left[\mathrm{\_Emden}\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{modified\_Emden\_ode}\right)$

$\left[\left[\mathrm{\_Emden}\,\mathrm{\_modified}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Emden\_Fowler\_ode}\right)$

$\left[\left[\mathrm{\_Emden}\,\mathrm{\_Fowler}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{symgen}\left(\mathrm{Emden\_ode}\,\mathrm{way}=3\right)$

$\left[\mathrm{\_\ξ}=-\frac{x\left(n-1\right)}{2}\,\mathrm{\_\η}=y\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Emden\_ode}\,\mathrm{HINT}=\left[-\frac{1}{2}xn+\frac{1}{2}x\,y\right]\right)$

$\mathrm{ans}≔y\left(x\right)=\left(\mathrm{\_a}{\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right)\mathbf{where}\left[\left\{\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=\left(\frac{{\mathrm{\_a}}^n{n}^2}{4}-\frac{{\mathrm{\_a}}^{n}n}{2}-\frac{\mathrm{\_a}n}{2}+\frac{{\mathrm{\_a}}^n}{4}+\frac{3\mathrm{\_a}}{2}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^3+\left(-\frac{n}{2}+\frac{5}{2}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^2\right\}\,\left\{\mathrm{\_a}=y\left(x\right)x^{\frac{2}{n-1}}\,\mathrm{\_b}\left(\mathrm{\_a}\right)=-\frac{2}{x^{\frac{2}{n-1}}\left(\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)xn-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2y\left(x\right)\right)}\right\}\,\left\{x={\ⅇ}^{-\frac{\left(\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}\right)n}{2}+\frac{\left(\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}\right)}{2}+\frac{\mathrm{c\_\_1}}{2}}\,y\left(x\right)=\mathrm{\_a}{\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right\}\right]$

$\mathrm{reduced\_ode}≔\mathrm{op}\left(\left[2\,2\,1\,1\right]\,\mathrm{ans}\right)$

$\mathrm{reduced\_ode}≔\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=\left(\frac{{\mathrm{\_a}}^n{n}^2}{4}-\frac{{\mathrm{\_a}}^{n}n}{2}-\frac{\mathrm{\_a}n}{2}+\frac{{\mathrm{\_a}}^n}{4}+\frac{3\mathrm{\_a}}{2}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^3+\left(-\frac{n}{2}+\frac{5}{2}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^2$

$\mathrm{odeadvisor}\left(\mathrm{reduced\_ode}\right)$

$\left[\mathrm{\_Abel}\right]$