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 = 0.484945 (sec), leaf count = 75
\[\left \{\left \{y(x)\to -\frac {a b+\sqrt {b^3 (x+c_2){}^2-b e^{2 c_1}}}{b}\right \},\left \{y(x)\to \frac {-a b+\sqrt {b^3 (x+c_2){}^2-b e^{2 c_1}}}{b}\right \}\right \}\]
Maple ✓
cpu = 0.413 (sec), leaf count = 53
\[\left [y \left (x \right ) = -a -\sqrt {b \,x^{2}-2 \textit {\_C1} x +a^{2}+2 \textit {\_C2}}, y \left (x \right ) = -a +\sqrt {b \,x^{2}-2 \textit {\_C1} x +a^{2}+2 \textit {\_C2}}\right ]\] 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))
Maple raw output
[y(x) = -a-(b*x^2-2*_C1*x+a^2+2*_C2)^(1/2), y(x) = -a+(b*x^2-2*_C1*x+a^2+2*_C2)^(1/2)]