4.19.8 \(x \left (x^3-2 y(x)\right ) y'(x)-\left (2 x^3-y(x)\right ) y(x)+x^2 y'(x)^2=0\)

ODE
\[ x \left (x^3-2 y(x)\right ) y'(x)-\left (2 x^3-y(x)\right ) y(x)+x^2 y'(x)^2=0 \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
Change of variable

Mathematica
cpu = 0.889102 (sec), leaf count = 154

\[\left \{\text {Solve}\left [\frac {1}{2} \left (-\frac {2 x^{5/2} \sqrt {\frac {x^3}{y(x)}+4} \sqrt {y(x)} \sinh ^{-1}\left (\frac {x^{3/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {x^8+4 x^5 y(x)}}+\log (y(x))-\log (x)\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {x^{5/2} \sqrt {\frac {x^3}{y(x)}+4} \sqrt {y(x)} \sinh ^{-1}\left (\frac {x^{3/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {x^8+4 x^5 y(x)}}+\frac {1}{2} \log (y(x))-\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}\]

Maple
cpu = 0.173 (sec), leaf count = 130

\[ \left \{ {\frac {1}{x} \left (\left (-{\it \_C1}\,x+y \relax (x ) \right ) \sqrt {{x}^{6}+4\,{x}^{3}y \relax (x ) }-{\it \_C1}\,{x}^{4}-{x}^{3}y \relax (x ) \right ) \left ({x}^{3}+\sqrt {{x}^{6}+4\,{x}^{3}y \relax (x ) } \right ) ^{-1}}=0,{\frac {1}{y \relax (x ) } \left (\left (-{\it \_C1}\,y \relax (x ) +x \right ) \sqrt {{x}^{6}+4\,{x}^{3}y \relax (x ) }-y \relax (x ) {\it \_C1}\,{x}^{3}-{x}^{4} \right ) \left ({x}^{3}+\sqrt {{x}^{6}+4\,{x}^{3}y \relax (x ) } \right ) ^{-1}}=0,y \relax (x ) =-{\frac {{x}^{3}}{4}} \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[(-Log[x] + Log[y[x]] - (2*x^(5/2)*ArcSinh[x^(3/2)/(2*Sqrt[y[x]])]*Sqrt[4 
+ x^3/y[x]]*Sqrt[y[x]])/Sqrt[x^8 + 4*x^5*y[x]])/2 == C[1], y[x]], Solve[-Log[x]/
2 + Log[y[x]]/2 + (x^(5/2)*ArcSinh[x^(3/2)/(2*Sqrt[y[x]])]*Sqrt[4 + x^3/y[x]]*Sq
rt[y[x]])/Sqrt[x^8 + 4*x^5*y[x]] == C[1], y[x]]}

Maple raw input

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

Maple raw output

y(x) = -1/4*x^3, ((-_C1*y(x)+x)*(x^6+4*x^3*y(x))^(1/2)-y(x)*_C1*x^3-x^4)/y(x)/(x
^3+(x^6+4*x^3*y(x))^(1/2)) = 0, ((-_C1*x+y(x))*(x^6+4*x^3*y(x))^(1/2)-_C1*x^4-x^
3*y(x))/(x^3+(x^6+4*x^3*y(x))^(1/2))/x = 0