##### 4.18.40 $$4 x y'(x)^2-3 y(x) y'(x)+3=0$$

ODE
$4 x y'(x)^2-3 y(x) y'(x)+3=0$ ODE Classiﬁcation

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

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 0.935606 (sec), leaf count = 187

$\left \{\left \{y(x)\to -\frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\right \},\left \{y(x)\to -\frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\right \}\right \}$

Maple
cpu = 0.228 (sec), leaf count = 153

$\left [y \left (x \right ) = -\frac {2 \sqrt {x \left (3+\sqrt {16 x \textit {\_C1} +9}\right )}}{3}-\frac {2 x}{\sqrt {x \left (3+\sqrt {16 x \textit {\_C1} +9}\right )}}, y \left (x \right ) = \frac {2 \sqrt {x \left (3+\sqrt {16 x \textit {\_C1} +9}\right )}}{3}+\frac {2 x}{\sqrt {x \left (3+\sqrt {16 x \textit {\_C1} +9}\right )}}, y \left (x \right ) = -\frac {2 \sqrt {-x \left (-3+\sqrt {16 x \textit {\_C1} +9}\right )}}{3}-\frac {2 x}{\sqrt {-x \left (-3+\sqrt {16 x \textit {\_C1} +9}\right )}}, y \left (x \right ) = \frac {2 \sqrt {-x \left (-3+\sqrt {16 x \textit {\_C1} +9}\right )}}{3}+\frac {2 x}{\sqrt {-x \left (-3+\sqrt {16 x \textit {\_C1} +9}\right )}}\right ]$ Mathematica raw input

DSolve[3 - 3*y[x]*y'[x] + 4*x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/6*Sqrt[E^C[1] - (E^C[1] - 144*x)^(3/2)/E^(C[1]/2) + 432*x]/Sqrt[3]}
, {y[x] -> Sqrt[E^C[1] - (E^C[1] - 144*x)^(3/2)/E^(C[1]/2) + 432*x]/(6*Sqrt[3])}
, {y[x] -> -1/6*Sqrt[E^C[1] + (E^C[1] - 144*x)^(3/2)/E^(C[1]/2) + 432*x]/Sqrt[3]
}, {y[x] -> Sqrt[E^C[1] + (E^C[1] - 144*x)^(3/2)/E^(C[1]/2) + 432*x]/(6*Sqrt[3])
}}

Maple raw input

dsolve(4*x*diff(y(x),x)^2-3*y(x)*diff(y(x),x)+3 = 0, y(x))

Maple raw output

[y(x) = -2/3*(x*(3+(16*_C1*x+9)^(1/2)))^(1/2)-2*x/(x*(3+(16*_C1*x+9)^(1/2)))^(1/
2), y(x) = 2/3*(x*(3+(16*_C1*x+9)^(1/2)))^(1/2)+2*x/(x*(3+(16*_C1*x+9)^(1/2)))^(
1/2), y(x) = -2/3*(-x*(-3+(16*_C1*x+9)^(1/2)))^(1/2)-2*x/(-x*(-3+(16*_C1*x+9)^(1
/2)))^(1/2), y(x) = 2/3*(-x*(-3+(16*_C1*x+9)^(1/2)))^(1/2)+2*x/(-x*(-3+(16*_C1*x
+9)^(1/2)))^(1/2)]