ODE
\[ y''(x)=A x^a y(x)^b y'(x)^c \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.0888083 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == A*x^a*y[x]^b*Derivative[1][y][x]^c, y[x], x]
Maple ✓
cpu = 2.429 (sec), leaf count = 206
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =-{\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}}{ \left ( a-c+2 \right ) ^{2}} \left ( A{{\it \_a}}^{b}{\it \_b} \left ( {\it \_a} \right ) \left ( b+c-1 \right ) ^{2} \left ( -{\frac { \left ( a-c+2 \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) {\it \_a}+1 \right ) }{{\it \_b} \left ( {\it \_a} \right ) \left ( b+c-1 \right ) }} \right ) ^{c}- \left ( {\it \_a}\, \left ( 1+a+b \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,a+b-c+3 \right ) \left ( a-c+2 \right ) \right ) } \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {a-c+2}{b+c-1}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {-a+c-2}{ \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) x \left ( b+c-1 \right ) +y \left ( x \right ) \left ( a-c+2 \right ) } \left ( {x}^{{\frac {a-c+2}{b+c-1}}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{-{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( b+c-1 \right ) }{a-c+2}}}},y \left ( x \right ) ={\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y''[x] == A*x^a*y[x]^b*y'[x]^c,y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == A*x^a*y[x]^b*Derivative[1][y][x]^c, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = A*x^a*y(x)^b*diff(y(x),x)^c, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a*exp(Int(_b(_a),_a)+_C1),[{diff(_b(_a),_a) = -(A*_a^b*_b(_a)*(b+c-1)^2*(-(a-c+2)*(_b(_a)*_a+1)/_b(_a)/(b+c-1))^c-(_a*(1+a+b)*_b(_a)+2*a+b-c+3)*(a-c+2))*_b(_a)^2/(a-c+2)^2}, {_a = y(x)*x^(1/(b+c-1)*(a-c+2)), _b(_a) = (-a+c-2)/(diff(y(x),x)*x*(b+c-1)+y(x)*(a-c+2))/(x^(1/(b+c-1)*(a-c+2)))}, {x = exp(-(Int(_b(_a),_a)+_C1)*(b+c-1)/(a-c+2)), y(x) = _a*exp(Int(_b(_a),_a)+_C1)}])