##### 4.16.48 $$y'(x)^2-(2 x+1) y'(x)-(1-x) x=0$$

ODE
$y'(x)^2-(2 x+1) y'(x)-(1-x) x=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $$y'$$

Mathematica
cpu = 0.181493 (sec), leaf count = 62

$\left \{\left \{y(x)\to \frac {x^2}{2}+\frac {x}{2}-\frac {1}{24} (8 x+1)^{3/2}+c_1\right \},\left \{y(x)\to \frac {1}{2} \left (x^2+x+\frac {1}{12} (8 x+1)^{3/2}\right )+c_1\right \}\right \}$

Maple
cpu = 0.033 (sec), leaf count = 45

$\left [y \left (x \right ) = \frac {x}{2}+\frac {x^{2}}{2}-\frac {\left (8 x +1\right )^{\frac {3}{2}}}{24}+\textit {\_C1}, y \left (x \right ) = \frac {x}{2}+\frac {x^{2}}{2}+\frac {\left (8 x +1\right )^{\frac {3}{2}}}{24}+\textit {\_C1}\right ]$ Mathematica raw input

DSolve[-((1 - x)*x) - (1 + 2*x)*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> x/2 + x^2/2 - (1 + 8*x)^(3/2)/24 + C[1]}, {y[x] -> (x + x^2 + (1 + 8*x
)^(3/2)/12)/2 + C[1]}}

Maple raw input

dsolve(diff(y(x),x)^2-(1+2*x)*diff(y(x),x)-x*(1-x) = 0, y(x))

Maple raw output

[y(x) = 1/2*x+1/2*x^2-1/24*(8*x+1)^(3/2)+_C1, y(x) = 1/2*x+1/2*x^2+1/24*(8*x+1)^
(3/2)+_C1]