ODE
\[ (y(x)+x) y''(x)+y'(x)^2-y'(x)=0 \] ODE Classification
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.984313 (sec), leaf count = 112
\[\left \{\left \{y(x)\to x-\frac {e^{-2 c_1} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}}{\sqrt {2}}+\frac {e^{-c_1}}{2}+2 c_2\right \},\left \{y(x)\to x+\frac {e^{-2 c_1} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}}{\sqrt {2}}+\frac {e^{-c_1}}{2}+2 c_2\right \}\right \}\]
Maple ✓
cpu = 0.502 (sec), leaf count = 16
\[\left [y \left (x \right ) = \sqrt {2 x +\textit {\_C1}}\, \textit {\_C2} +\textit {\_C1} +x\right ]\] Mathematica raw input
DSolve[-y'[x] + y'[x]^2 + (x + y[x])*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> 1/(2*E^C[1]) + x + 2*C[2] - Sqrt[E^(2*C[1])*(1 + 4*E^C[1]*(x + C[2]))]/(Sqrt[2]*E^(2*C[1]))}, {y[x] -> 1/(2*E^C[1]) + x + 2*C[2] + Sqrt[E^(2*C[1])*(1 + 4*E^C[1]*(x + C[2]))]/(Sqrt[2]*E^(2*C[1]))}}
Maple raw input
dsolve((x+y(x))*diff(diff(y(x),x),x)+diff(y(x),x)^2-diff(y(x),x) = 0, y(x))
Maple raw output
[y(x) = (2*x+_C1)^(1/2)*_C2+_C1+x]