4.18.23 \(9 x^2+x y'(x)^2-3 y(x) y'(x)=0\)

ODE
\[ 9 x^2+x y'(x)^2-3 y(x) y'(x)=0 \] ODE Classification

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 1.82809 (sec), leaf count = 357

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

Maple
cpu = 0.096 (sec), leaf count = 61

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-4\,{x}^{3}=0,{\frac {y \relax (x ) }{{x}^{3}}}+{\frac {1}{{x}^{3}}\sqrt { \left (y \relax (x ) \right ) ^{2}-4\,{x}^{3}}}-{\it \_C1}=0,y \relax (x ) +\sqrt { \left (y \relax (x ) \right ) ^{2}-4\,{x}^{3}}-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[9*x^2 - 3*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*diff(y(x),x)^2-3*y(x)*diff(y(x),x)+9*x^2 = 0, y(x),'implicit')

Maple raw output

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