4.19.8 $x(x^3-2y(x))y^{\prime }(x)-(2x^3-y(x))y(x)+x^2{y}^{\prime }(x{)}^2=0$

ODE
$$x(x^3-2y(x))y^{\prime }(x)-(2x^3-y(x))y(x)+x^2{y}^{\prime }(x{)}^2=0$$ ODE Classification

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

Book solution method
Change of variable

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

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

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

$$\{\frac{1}{x}((-\mathit{\_}\mathit{C}\mathit{1}x+y\left(x\right))\sqrt{x^6+4x^{3}y\left(x\right)}-\mathit{\_}\mathit{C}\mathit{1}{x}^4-x^{3}y\left(x\right)){(x^3+\sqrt{x^6+4x^{3}y\left(x\right)})}^{-1}=0,\frac{1}{y\left(x\right)}((-\mathit{\_}\mathit{C}\mathit{1}y\left(x\right)+x)\sqrt{x^6+4x^{3}y\left(x\right)}-y\left(x\right)\mathit{\_}\mathit{C}\mathit{1}{x}^3-x^4){(x^3+\sqrt{x^6+4x^{3}y\left(x\right)})}^{-1}=0,y\left(x\right)=-\frac{x^3}{4}\}$$ 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