##### 4.20.26 $$y(x)^2 y'(x)^2-3 x y'(x)+y(x)=0$$

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

[[_1st_order, _with_linear_symmetries], _rational]

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

Mathematica
cpu = 0.415163 (sec), leaf count = 185

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

Maple
cpu = 1.577 (sec), leaf count = 120

$\left [y \left (x \right ) = \frac {18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2}, y \left (x \right ) = -\frac {18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}, y \left (x \right ) = -\frac {18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-6 \textit {\_a}^{3}+\frac {9 \sqrt {-4 \textit {\_a}^{3}+9}}{2}+\frac {27}{2}}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} +\textit {\_C1} \right ) x^{\frac {2}{3}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> E^(C[1]/3)*(E^C[1] - (3*I)*x)^(1/3)}, {y[x] -> -((-1)^(1/3)*E^(C[1]/3)
*(E^C[1] - (3*I)*x)^(1/3))}, {y[x] -> (-1)^(2/3)*E^(C[1]/3)*(E^C[1] - (3*I)*x)^(
1/3)}, {y[x] -> E^(C[1]/3)*(E^C[1] + (3*I)*x)^(1/3)}, {y[x] -> -((-1)^(1/3)*E^(C
[1]/3)*(E^C[1] + (3*I)*x)^(1/3))}, {y[x] -> (-1)^(2/3)*E^(C[1]/3)*(E^C[1] + (3*I
)*x)^(1/3)}}

Maple raw input

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

Maple raw output

[y(x) = 1/2*18^(1/3)*(x^2)^(1/3), y(x) = -1/4*18^(1/3)*(x^2)^(1/3)-1/4*I*3^(1/2)
*18^(1/3)*(x^2)^(1/3), y(x) = -1/4*18^(1/3)*(x^2)^(1/3)+1/4*I*3^(1/2)*18^(1/3)*(
x^2)^(1/3), y(x) = RootOf(-ln(x)+Intat(3/2*(-4*_a^3+3*(-4*_a^3+9)^(1/2)+9)/_a/(4
*_a^3-9),_a = _Z)+_C1)*x^(2/3)]