ODE
\[ y''(x)=f\left (a x+b y(x),y'(x)\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.132383 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == f[a*x + b*y[x], Derivative[1][y][x]], y[x], x]
Maple ✓
cpu = 0.132 (sec), leaf count = 115
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( -{\frac {a \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) -{\it \_a}\,b}{b}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =-f \left ( {\it \_a}\,b,{\frac {-a{\it \_b} \left ( {\it \_a} \right ) +b}{{\it \_b} \left ( {\it \_a} \right ) b}} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3} \right \} , \left \{ {\it \_a}={\frac {ax+by \left ( x \right ) }{b}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {b}{a+b{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }} \right \} , \left \{ x=\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1},y \left ( x \right ) =-{\frac {a \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) -{\it \_a}\,b}{b}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y''[x] == f[a*x + b*y[x], y'[x]],y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == f[a*x + b*y[x], Derivative[1][y][x]], y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = f(a*x+b*y(x),diff(y(x),x)), y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(-(a*(Int(_b(_a),_a)+_C1)-_a*b)/b,[{diff(_b(_a),_a) = -f(_a*b,(-a*_b(_a)+b)/b/_b(_a))*_b(_a)^3}, {_a = (a*x+b*y(x))/b, _b(_a) = 1/(a+b*diff(y(x),x))*b}, {x = Int(_b(_a),_a)+_C1, y(x) = -(a*(Int(_b(_a),_a)+_C1)-_a*b)/b}])