##### 4.16.37 $$y'(x)^2-x y'(x)-y(x)=0$$

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 0.451164 (sec), leaf count = 1493

$\left \{\left \{y(x)\to \frac {x^2}{2}+\frac {\left (x^3+8 \cosh (3 c_1)+8 \sinh (3 c_1)\right ) x}{4 \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {1}{4} \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^3+8 \cosh (3 c_1)+8 \sinh (3 c_1)\right ) x}{8 \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {1}{8} i \left (i+\sqrt {3}\right ) \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {i \left (i+\sqrt {3}\right ) \left (x^3+8 \cosh (3 c_1)+8 \sinh (3 c_1)\right ) x}{8 \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}-\frac {1}{8} i \left (-i+\sqrt {3}\right ) \sqrt [3]{-x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3+8 \cosh (6 c_1)+8 \sinh (6 c_1)+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {\left (x^3-2 \cosh (3 c_1)-2 \sinh (3 c_1)\right ) x}{2\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {\sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{4 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {x^2}{2}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^3-2 \cosh (3 c_1)-2 \sinh (3 c_1)\right ) x}{4\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {i \left (i+\sqrt {3}\right ) \sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{8 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {i \left (i+\sqrt {3}\right ) \left (x^3-2 \cosh (3 c_1)-2 \sinh (3 c_1)\right ) x}{4\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}-\frac {i \left (-i+\sqrt {3}\right ) \sqrt [3]{-2 x^6-10 \cosh (3 c_1) x^3-10 \sinh (3 c_1) x^3+\cosh (6 c_1)+\sinh (6 c_1)+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{8 \sqrt [3]{2}}\right \}\right \}$

Maple
cpu = 0.044 (sec), leaf count = 77

$\left [\frac {\textit {\_C1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y \left (x \right )}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}+4 y \left (x \right )}}{3} = 0, \frac {\textit {\_C1}}{\sqrt {-2 \sqrt {x^{2}+4 y \left (x \right )}+2 x}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{3} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> x^2/2 + (x*(x^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]]))/(4*(-x^6 + 20*x^3*
Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(
((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2]
+ Sinh[(15*C[1])/2]))])^(1/3)) + (-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] +
20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (
1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3)/4}
, {y[x] -> x^2/2 - ((I/8)*(-I + Sqrt[3])*x*(x^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]
]))/(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[
6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*
(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3) + (I/8)*(I + Sqrt[3])*(-x^6 + 2
0*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*S
qrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1
])/2] + Sinh[(15*C[1])/2]))])^(1/3)}, {y[x] -> x^2/2 + ((I/8)*(I + Sqrt[3])*x*(x
^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]]))/(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[
1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/
2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1
/3) - (I/8)*(-I + Sqrt[3])*(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3
*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^
3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3)}, {y[x]
-> x^2/2 + (x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C[1]]))/(2*2^(2/3)*(-2*x^6 - 10*x
^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 +
 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] +
Sinh[(15*C[1])/2])])^(1/3)) + (-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*
x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^
3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3)/(4*2^(1/3
))}, {y[x] -> x^2/2 - ((I/4)*(-I + Sqrt[3])*x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C
[1]]))/(2^(2/3)*(-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1
]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1
])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3)) + ((I/8)*(I + Sqrt[3])
*(-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1
]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[
(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3))/2^(1/3)}, {y[x] -> x^2/2 + ((I/4)*(I
+ Sqrt[3])*x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C[1]]))/(2^(2/3)*(-2*x^6 - 10*x^3*
Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*
x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sin
h[(15*C[1])/2])])^(1/3)) - ((I/8)*(-I + Sqrt[3])*(-2*x^6 - 10*x^3*Cosh[3*C[1]] +
 Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C
[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]
)])^(1/3))/2^(1/3)}}

Maple raw input

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

Maple raw output

[1/(2*x+2*(x^2+4*y(x))^(1/2))^(1/2)*_C1+2/3*x-1/3*(x^2+4*y(x))^(1/2) = 0, 1/(-2*
(x^2+4*y(x))^(1/2)+2*x)^(1/2)*_C1+2/3*x+1/3*(x^2+4*y(x))^(1/2) = 0]