ODE
\[ (a+y(x)) y''(x)+b y'(x)^2=0 \] ODE Classification
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0494846 (sec), leaf count = 25
\[\left \{\left \{y(x)\to \left ((b+1) c_1 \left (c_2+x\right )\right ){}^{\frac {1}{b+1}}-a\right \}\right \}\]
Maple ✓
cpu = 0.018 (sec), leaf count = 27
\[ \left \{ {\frac { \left ( a+y \left ( x \right ) \right ) \left ( a+y \left ( x \right ) \right ) ^{b}}{b+1}}-{\it \_C1}\,x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[b*y'[x]^2 + (a + y[x])*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -a + ((1 + b)*C[1]*(x + C[2]))^(1 + b)^(-1)}}
Maple raw input
dsolve((a+y(x))*diff(diff(y(x),x),x)+b*diff(y(x),x)^2 = 0, y(x),'implicit')
Maple raw output
(a+y(x))/(b+1)*(a+y(x))^b-_C1*x-_C2 = 0