ODE
\[ -2 x^2 y'(x)+2 y'(x)^2+3 x y(x)=0 \] ODE Classification
[[_homogeneous, `class G`]]
Book solution method
Homogeneous ODE, The Isobaric equation
Mathematica ✓
cpu = 0.667712 (sec), leaf count = 127
\[\left \{\text {Solve}\left [\frac {2 \sqrt {x^4-6 x y(x)} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-6 y(x)}}\right )}{\sqrt {x} \sqrt {x^3-6 y(x)}}+3 c_1=\log (y(x)),y(x)\right ],\text {Solve}\left [3 c_1=\frac {2 \sqrt {x^4-6 x y(x)} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-6 y(x)}}\right )}{\sqrt {x} \sqrt {x^3-6 y(x)}}+\log (y(x)),y(x)\right ]\right \}\]
Maple ✓
cpu = 0.502 (sec), leaf count = 123
\[\left [y \left (x \right ) = \frac {x^{3}}{6}, y \left (x \right ) = -\frac {-\textit {\_C1} \,x^{3}-x \left (-x^{2} \textit {\_C1} +\sqrt {-6 x \textit {\_C1}}\right )-3}{3 \textit {\_C1}}, y \left (x \right ) = -\frac {-\textit {\_C1} \,x^{3}+x \left (x^{2} \textit {\_C1} +\sqrt {-6 x \textit {\_C1}}\right )-3}{3 \textit {\_C1}}, y \left (x \right ) = \frac {x^{3}}{3}+\frac {\left (-x^{2}-\sqrt {-6 x \textit {\_C1}}\right ) x}{3}+\textit {\_C1}, y \left (x \right ) = \frac {x^{3}}{3}+\frac {\left (-x^{2}+\sqrt {-6 x \textit {\_C1}}\right ) x}{3}+\textit {\_C1}\right ]\] Mathematica raw input
DSolve[3*x*y[x] - 2*x^2*y'[x] + 2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[3*C[1] + (2*ArcTanh[x^(3/2)/Sqrt[x^3 - 6*y[x]]]*Sqrt[x^4 - 6*x*y[x]])/(Sqrt[x]*Sqrt[x^3 - 6*y[x]]) == Log[y[x]], y[x]], Solve[3*C[1] == Log[y[x]] + (2*ArcTanh[x^(3/2)/Sqrt[x^3 - 6*y[x]]]*Sqrt[x^4 - 6*x*y[x]])/(Sqrt[x]*Sqrt[x^3 - 6*y[x]]), y[x]]}
Maple raw input
dsolve(2*diff(y(x),x)^2-2*x^2*diff(y(x),x)+3*x*y(x) = 0, y(x))
Maple raw output
[y(x) = 1/6*x^3, y(x) = -1/3*(-_C1*x^3-x*(-x^2*_C1+(-6*x*_C1)^(1/2))-3)/_C1, y(x) = -1/3*(-_C1*x^3+x*(x^2*_C1+(-6*x*_C1)^(1/2))-3)/_C1, y(x) = 1/3*x^3+1/3*(-x^2-(-6*x*_C1)^(1/2))*x+_C1, y(x) = 1/3*x^3+1/3*(-x^2+(-6*x*_C1)^(1/2))*x+_C1]