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 = 0.327781 (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 = 0.016 (sec), leaf count = 30

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {2\,{{\it \_T}}^{3}+{\it \_C1}}{{{\it \_T}}^{2}}},y \left ({\it \_T} \right ) ={\frac {{{\it \_T}}^{3}+2\,{\it \_C1}}{{\it \_T}}}] \right \} \] 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),'implicit')

Maple raw output

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