##### 4.19.37 $$x^4 y'(x)^2-x y'(x)-y(x)=0$$

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

[[_homogeneous, class G], _rational]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 0.591167 (sec), leaf count = 118

$\left \{\text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}+\frac {1}{2} \log (y(x))+c_1=0,y(x)\right ],\text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}=\frac {1}{2} \log (y(x))+c_1,y(x)\right ]\right \}$

Maple
cpu = 7.536 (sec), leaf count = 135

$\left [y \left (x \right ) = -\frac {1}{4 x^{2}}, y \left (x \right ) = \frac {-\textit {\_C1} \left (2 i x -\textit {\_C1} \right )-\textit {\_C1}^{2}-2 x^{2}}{2 \textit {\_C1}^{2} x^{2}}, y \left (x \right ) = \frac {-\textit {\_C1} \left (-2 i x -\textit {\_C1} \right )-\textit {\_C1}^{2}-2 x^{2}}{2 \textit {\_C1}^{2} x^{2}}, y \left (x \right ) = \frac {\textit {\_C1} \left (2 i x +\textit {\_C1} \right )-2 x^{2}-\textit {\_C1}^{2}}{2 \textit {\_C1}^{2} x^{2}}, y \left (x \right ) = \frac {\textit {\_C1} \left (-2 i x +\textit {\_C1} \right )-2 x^{2}-\textit {\_C1}^{2}}{2 \textit {\_C1}^{2} x^{2}}\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[C[1] + Log[y[x]]/2 + (x*ArcTanh[Sqrt[1 + 4*x^2*y[x]]]*Sqrt[1 + 4*x^2*y[x]
])/Sqrt[x^2 + 4*x^4*y[x]] == 0, y[x]], Solve[(x*ArcTanh[Sqrt[1 + 4*x^2*y[x]]]*Sq
rt[1 + 4*x^2*y[x]])/Sqrt[x^2 + 4*x^4*y[x]] == C[1] + Log[y[x]]/2, y[x]]}

Maple raw input

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

Maple raw output

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