4.18.23 $9x^2+x{y}^{\prime }(x{)}^2-3y(x)y^{\prime }(x)=0$

ODE
$$9x^2+x{y}^{\prime }(x{)}^2-3y(x)y^{\prime }(x)=0$$ ODE Classification

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for $y$

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

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

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

$$\{{\left(y\left(x\right)\right)}^2-4x^3=0,\frac{y\left(x\right)}{x^3}+\frac{1}{x^3}\sqrt{{\left(y\left(x\right)\right)}^2-4x^3}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4x^3}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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