ODE
\[ x^a y''(x)+y(x)^b=0 \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.0445981 (sec), leaf count = 0 , could not solve
DSolve[y[x]^b + x^a*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 4.556 (sec), leaf count = 158
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) ={\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \left ( b-1 \right ) ^{2}{{\it \_a}}^{b}+ \left ( {\it \_a}\, \left ( a-1-b \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,a-b-3 \right ) \left ( a-2 \right ) \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}}{ \left ( a-2 \right ) ^{2}}} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {2-a}{b-1}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {2-a}{-x \left ( b-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) \left ( a-2 \right ) } \left ( {x}^{{\frac {2-a}{b-1}}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( b-1 \right ) }{a-2}}}},y \left ( x \right ) ={\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y[x]^b + x^a*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[y[x]^b + x^a*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(x^a*diff(diff(y(x),x),x)+y(x)^b = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a*exp(Int(_b(_a),_a)+_C1),[{diff(_b(_a),_a) = (_b(_a)*(b-1)^2*_a^b+(_a*(a-1-b)*_b(_a)+2*a-b-3)*(a-2))*_b(_a)^2/(a-2)^2}, {_a = y(x)*x^((2-a)/(b-1)), _b(_a) = (2-a)/(x^((2-a)/(b-1)))/(-x*(b-1)*diff(y(x),x)+y(x)*(a-2))}, {x = exp((Int(_b(_a),_a)+_C1)*(b-1)/(a-2)), y(x) = _a*exp(Int(_b(_a),_a)+_C1)}])