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 = 1.0571 (sec), leaf count = 406

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

Maple
cpu = 0.059 (sec), leaf count = 69

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {\ln \left ({x}^{2}y \relax (x ) \right ) }{2}}-{\it Artanh} \left (\sqrt {4\,{x}^{2}y \relax (x ) +1} \right ) =0,\ln \relax (x ) -{\it \_C1}-{\frac {\ln \left ({x}^{2}y \relax (x ) \right ) }{2}}+{\it Artanh} \left (\sqrt {4\,{x}^{2}y \relax (x ) +1} \right ) =0,y \relax (x ) =-{\frac {1}{4\,{x}^{2}}} \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[(2*x*Log[x]*Sqrt[1 + 4*x^2*y[x]] + x*Log[y[x]]*Sqrt[1 + 4*x^2*y[x]] - x*L
og[1 + 4*x^2*y[x]]*Sqrt[1 + 4*x^2*y[x]] + x*Log[x + 4*x^3*y[x]]*Sqrt[1 + 4*x^2*y
[x]] - 2*x*Log[1 + Sqrt[1 + 4*x^2*y[x]]]*Sqrt[1 + 4*x^2*y[x]] + Log[1 + 1/(4*x^2
*y[x])]*Sqrt[x^2 + 4*x^4*y[x]] - Log[1 + 4*x^2*y[x]]*Sqrt[x^2 + 4*x^4*y[x]])/(2*
Sqrt[x^2 + 4*x^4*y[x]]) == C[1], y[x]], Solve[(-2*x*Log[x]*Sqrt[1 + 4*x^2*y[x]] 
- x*Log[y[x]]*Sqrt[1 + 4*x^2*y[x]] + x*Log[1 + 4*x^2*y[x]]*Sqrt[1 + 4*x^2*y[x]] 
- x*Log[x + 4*x^3*y[x]]*Sqrt[1 + 4*x^2*y[x]] + 2*x*Log[1 + Sqrt[1 + 4*x^2*y[x]]]
*Sqrt[1 + 4*x^2*y[x]] + Log[1 + 1/(4*x^2*y[x])]*Sqrt[x^2 + 4*x^4*y[x]] - Log[1 +
 4*x^2*y[x]]*Sqrt[x^2 + 4*x^4*y[x]])/(2*Sqrt[x^2 + 4*x^4*y[x]]) == C[1], y[x]]}

Maple raw input

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

Maple raw output

y(x) = -1/4/x^2, ln(x)-_C1-1/2*ln(x^2*y(x))-arctanh((4*x^2*y(x)+1)^(1/2)) = 0, l
n(x)-_C1-1/2*ln(x^2*y(x))+arctanh((4*x^2*y(x)+1)^(1/2)) = 0