##### 4.18.19 $$a+x y'(x)^2-2 y(x) y'(x)=0$$

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

[[_homogeneous, class G], _rational, _dAlembert]

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

Mathematica
cpu = 0.483925 (sec), leaf count = 773

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

Maple
cpu = 0.143 (sec), leaf count = 897

$\left [y \left (x \right ) = \frac {x \left (\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}+\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {x}{3 \textit {\_C1}}\right )}{2}+\frac {a}{\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{3 \textit {\_C1}}+\frac {4 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {2 x}{3 \textit {\_C1}}}, y \left (x \right ) = \frac {x \left (-\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {x}{3 \textit {\_C1}}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}\right )}{2}\right )}{2}+\frac {a}{-\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {2 x}{3 \textit {\_C1}}-i \sqrt {3}\, \left (\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}\right )}, y \left (x \right ) = \frac {x \left (-\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {x}{3 \textit {\_C1}}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}\right )}{2}\right )}{2}+\frac {a}{-\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {2 x}{3 \textit {\_C1}}+i \sqrt {3}\, \left (\frac {\left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2 x^{2}}{3 \textit {\_C1} \left (-36 a \,\textit {\_C1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,\textit {\_C1}^{2}-4 x^{3}\right )}\, \textit {\_C1} \right )^{\frac {1}{3}}}\right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (8*a*E^(3*C[1])*x + a^4*x^4 + a^2*x^2*(-8*E^(6*C[1]) - 20*a^3*E^(3*C[1
])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])^(1/3) + (-8*E^(6
*C[1]) - 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x
^3)^3])^(2/3))/(4*E^((3*C[1])/2)*(-8*E^(6*C[1]) - 20*a^3*E^(3*C[1])*x^3 + a^6*x^
6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])^(1/3))}, {y[x] -> ((-8*I)*(-I +
 Sqrt[3])*a*E^(3*C[1])*x + (-1 - I*Sqrt[3])*a^4*x^4 + 2*a^2*x^2*(-8*E^(6*C[1]) -
 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])^
(1/3) + I*(I + Sqrt[3])*(-8*E^(6*C[1]) - 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqr
t[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])^(2/3))/(8*E^((3*C[1])/2)*(-8*E^(6*C[1])
- 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])
^(1/3))}, {y[x] -> ((8*I)*(I + Sqrt[3])*a*E^(3*C[1])*x + I*(I + Sqrt[3])*a^4*x^4
 + 2*a^2*x^2*(-8*E^(6*C[1]) - 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1]
)*(E^(3*C[1]) - a^3*x^3)^3])^(1/3) - I*(-I + Sqrt[3])*(-8*E^(6*C[1]) - 20*a^3*E^
(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[1])*(E^(3*C[1]) - a^3*x^3)^3])^(2/3))/(8*
E^((3*C[1])/2)*(-8*E^(6*C[1]) - 20*a^3*E^(3*C[1])*x^3 + a^6*x^6 + 8*Sqrt[E^(3*C[
1])*(E^(3*C[1]) - a^3*x^3)^3])^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/2*x*(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)^(1
/3)+2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)^(1/3)+1/3
*x/_C1)+1/2*a/(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)^(1
/3)+2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)^(1/3)+1/3
*x/_C1), y(x) = 1/2*x*(-1/12/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/
2)*_C1)^(1/3)-1/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)
^(1/3)+1/3*x/_C1-1/2*I*3^(1/2)*(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^
3))^(1/2)*_C1)^(1/3)-2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/
2)*_C1)^(1/3)))+1/2*a/(-1/12/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/
2)*_C1)^(1/3)-1/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/2)*_C1)
^(1/3)+1/3*x/_C1-1/2*I*3^(1/2)*(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^
3))^(1/2)*_C1)^(1/3)-2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1/
2)*_C1)^(1/3))), y(x) = 1/2*x*(-1/12/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x
^3))^(1/2)*_C1)^(1/3)-1/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1
/2)*_C1)^(1/3)+1/3*x/_C1+1/2*I*3^(1/2)*(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^
2*a-4*x^3))^(1/2)*_C1)^(1/3)-2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x
^3))^(1/2)*_C1)^(1/3)))+1/2*a/(-1/12/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x
^3))^(1/2)*_C1)^(1/3)-1/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x^3))^(1
/2)*_C1)^(1/3)+1/3*x/_C1+1/2*I*3^(1/2)*(1/6/_C1*(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^
2*a-4*x^3))^(1/2)*_C1)^(1/3)-2/3*x^2/_C1/(-36*a*_C1^2+8*x^3+12*(a*(9*_C1^2*a-4*x
^3))^(1/2)*_C1)^(1/3)))]