##### 4.17.39 $$3 y'(x)^2-2 x y'(x)+y(x)=0$$

ODE
$3 y'(x)^2-2 x y'(x)+y(x)=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 0.519919 (sec), leaf count = 1093

$\left \{\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,6\right ]\right \}\right \}$

Maple
cpu = 0.098 (sec), leaf count = 656

$\left [y \left (x \right ) = -3 \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}\right )^{2}+2 x \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}\right ), y \left (x \right ) = -3 \left (-\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ), y \left (x \right ) = -3 \left (-\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 \textit {\_C1} +x^{3}+6 \sqrt {-3 \textit {\_C1} \,x^{3}+81 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )\right ]$ Mathematica raw input

DSolve[y[x] - 2*x*y'[x] + 3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[243*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378
*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 3*x^4*#1^4 - 24*x^2*#1^5 + 48*#1^6
& , 1]}, {y[x] -> Root[243*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*
#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 3*x^4*#1^4 - 24*x^2*#1^5 +
48*#1^6 & , 2]}, {y[x] -> Root[243*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[
1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 3*x^4*#1^4 - 24*x^2
*#1^5 + 48*#1^6 & , 3]}, {y[x] -> Root[243*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144
*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 3*x^4*#1^4
- 24*x^2*#1^5 + 48*#1^6 & , 4]}, {y[x] -> Root[243*E^(12*C[1]) - 16*E^(6*C[1])*x
^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 3*x
^4*#1^4 - 24*x^2*#1^5 + 48*#1^6 & , 5]}, {y[x] -> Root[243*E^(12*C[1]) - 16*E^(6
*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1
^3 + 3*x^4*#1^4 - 24*x^2*#1^5 + 48*#1^6 & , 6]}, {y[x] -> Root[3*E^(12*C[1]) - 1
6*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[
1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 1]}, {y[x] -> Root[3*E^(
12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 +
 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 2]}, {y[x] -
> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1]
)*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & ,
3]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 3
78*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 38
88*#1^6 & , 4]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1]
)*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x
^2*#1^5 + 3888*#1^6 & , 5]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 1
44*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#
1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 6]}}

Maple raw input

dsolve(3*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = -3*(1/6*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)+1/6*x^2/(-54*_
C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)+1/6*x)^2+2*x*(1/6*(-54*_C1+x^3+6*(-3
*_C1*x^3+81*_C1^2)^(1/2))^(1/3)+1/6*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/
2))^(1/3)+1/6*x), y(x) = -3*(-1/12*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(
1/3)-1/12*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)+1/6*x-1/2*I*3^(1
/2)*(1/6*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)-1/6*x^2/(-54*_C1+x^3+
6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)))^2+2*x*(-1/12*(-54*_C1+x^3+6*(-3*_C1*x^3+8
1*_C1^2)^(1/2))^(1/3)-1/12*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)
+1/6*x-1/2*I*3^(1/2)*(1/6*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)-1/6*
x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3))), y(x) = -3*(-1/12*(-54*_
C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)-1/12*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+
81*_C1^2)^(1/2))^(1/3)+1/6*x+1/2*I*3^(1/2)*(1/6*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C
1^2)^(1/2))^(1/3)-1/6*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)))^2+
2*x*(-1/12*(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)-1/12*x^2/(-54*_C1+x
^3+6*(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)+1/6*x+1/2*I*3^(1/2)*(1/6*(-54*_C1+x^3+6*
(-3*_C1*x^3+81*_C1^2)^(1/2))^(1/3)-1/6*x^2/(-54*_C1+x^3+6*(-3*_C1*x^3+81*_C1^2)^
(1/2))^(1/3)))]