ODE
\[ a x^r y(x)^s+y''(x)=0 \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.0366374 (sec), leaf count = 0 , could not solve
DSolve[a*x^r*y[x]^s + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 3.106 (sec), leaf count = 151
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) ={\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( a{\it \_b} \left ( {\it \_a} \right ) \left ( s-1 \right ) ^{2}{{\it \_a}}^{s}+ \left ( {\it \_a}\, \left ( r+1+s \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,r+s+3 \right ) \left ( r+2 \right ) \right ) }{ \left ( r+2 \right ) ^{2}}} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {r+2}{s-1}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {-r-2}{x \left ( s-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) \left ( r+2 \right ) } \left ( {x}^{{\frac {r+2}{s-1}}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{-{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( s-1 \right ) }{r+2}}}},y \left ( x \right ) ={\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[a*x^r*y[x]^s + y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[a*x^r*y[x]^s + Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*x^r*y(x)^s = 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)*(s-1)^2*_a^s+(_a*(r+1+s)*_b(_a)+2*r+s+3)*(r+2))/(r+2)^2}, {_a = y(x)*x^(1/(s-1)*(r+2)), _b(_a) = (-r-2)/(x*(s-1)*diff(y(x),x)+y(x)*(r+2))/(x^(1/(s-1)*(r+2)))}, {x = exp(-(Int(_b(_a),_a)+_C1)*(s-1)/(r+2)), y(x) = _a*exp(Int(_b(_a),_a)+_C1)}])