ODE
\[ a x^m y(x)^n+x y''(x)+2 y'(x)=0 \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.979485 (sec), leaf count = 0 , could not solve
DSolve[a*x^m*y[x]^n + 2*Derivative[1][y][x] + x*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 4.763 (sec), leaf count = 155
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) ={\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( a{\it \_b} \left ( {\it \_a} \right ) \left ( n-1 \right ) ^{2}{{\it \_a}}^{n}+ \left ( {\it \_a}\, \left ( m-n+2 \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,m-n+3 \right ) \left ( m+1 \right ) \right ) }{ \left ( m+1 \right ) ^{2}}} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {m+1}{n-1}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {-m-1}{ \left ( n-1 \right ) x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) \left ( m+1 \right ) } \left ( {x}^{{\frac {m+1}{n-1}}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{-{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( n-1 \right ) }{m+1}}}},y \left ( x \right ) ={\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[a*x^m*y[x]^n + 2*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[a*x^m*y[x]^n + 2*Derivative[1][y][x] + x*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x)+a*x^m*y(x)^n = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a*exp(Int(_b(_a),_a)+_C1),[{diff(_b(_a),_a) = _b(_a)^2*(a*_b(_a)*(n-1)^2*_a^n+(_a*(m-n+2)*_b(_a)+2*m-n+3)*(m+1))/(m+1)^2}, {_a = y(x)*x^(1/(n-1)*(m+1)), _b(_a) = (-m-1)/(x^(1/(n-1)*(m+1)))/((n-1)*x*diff(y(x),x)+y(x)*(m+1))}, {x = exp(-(Int(_b(_a),_a)+_C1)*(n-1)/(m+1)), y(x) = _a*exp(Int(_b(_a),_a)+_C1)}])