ODE
\[ (a+y(x)) y''(x)+y'(x)^2=b \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.138605 (sec), leaf count = 75
\[\left \{\left \{y(x)\to -\frac {a b+\sqrt {b^3 \left (c_2+x\right ){}^2-b e^{2 c_1}}}{b}\right \},\left \{y(x)\to \frac {\sqrt {b^3 \left (c_2+x\right ){}^2-b e^{2 c_1}}-a b}{b}\right \}\right \}\]
Maple ✓
cpu = 0.04 (sec), leaf count = 26
\[ \left \{ {\frac {b{x}^{2}}{2}}-{\it \_C1}\,x-ay \left ( x \right ) -{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2}}+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y'[x]^2 + (a + y[x])*y''[x] == b,y[x],x]
Mathematica raw output
{{y[x] -> -((a*b + Sqrt[-(b*E^(2*C[1])) + b^3*(x + C[2])^2])/b)}, {y[x] -> (-(a*b) + Sqrt[-(b*E^(2*C[1])) + b^3*(x + C[2])^2])/b}}
Maple raw input
dsolve((a+y(x))*diff(diff(y(x),x),x)+diff(y(x),x)^2 = b, y(x),'implicit')
Maple raw output
1/2*b*x^2-_C1*x-a*y(x)-1/2*y(x)^2+_C2 = 0