##### 4.17.10 $$4 x^5 y'(x)-12 x^4 y(x)+y'(x)^2=0$$

ODE
$4 x^5 y'(x)-12 x^4 y(x)+y'(x)^2=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

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

Mathematica
cpu = 0.618029 (sec), leaf count = 125

$\left \{\text {Solve}\left [\frac {2 x^2 \sqrt {x^6+3 y(x)} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6+3 y(x)}}\right )}{\sqrt {x^4 \left (x^6+3 y(x)\right )}}+6 c_1=\log (y(x)),y(x)\right ],\text {Solve}\left [6 c_1=\frac {2 x^2 \sqrt {x^6+3 y(x)} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6+3 y(x)}}\right )}{\sqrt {x^4 \left (x^6+3 y(x)\right )}}+\log (y(x)),y(x)\right ]\right \}$

Maple
cpu = 1.279 (sec), leaf count = 23

$\left [y \left (x \right ) = -\frac {x^{6}}{3}, y \left (x \right ) = \textit {\_C1} \,x^{3}+\frac {3}{4} \textit {\_C1}^{2}\right ]$ Mathematica raw input

DSolve[-12*x^4*y[x] + 4*x^5*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[6*C[1] + (2*x^2*ArcTanh[x^3/Sqrt[x^6 + 3*y[x]]]*Sqrt[x^6 + 3*y[x]])/Sqrt[
x^4*(x^6 + 3*y[x])] == Log[y[x]], y[x]], Solve[6*C[1] == Log[y[x]] + (2*x^2*ArcT
anh[x^3/Sqrt[x^6 + 3*y[x]]]*Sqrt[x^6 + 3*y[x]])/Sqrt[x^4*(x^6 + 3*y[x])], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2+4*x^5*diff(y(x),x)-12*x^4*y(x) = 0, y(x))

Maple raw output

[y(x) = -1/3*x^6, y(x) = _C1*x^3+3/4*_C1^2]