4.19.44 $x^8{y}^{\prime }(x{)}^2+3x{y}^{\prime }(x)+9y(x)=0$

ODE
$$x^8{y}^{\prime }(x{)}^2+3x{y}^{\prime }(x)+9y(x)=0$$ ODE Classification

[[_homogeneous, `class G`]]

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

Mathematica ✓
cpu = 0.415011 (sec), leaf count = 120

$$\{\text{Solve}[6c_1+\frac{2\sqrt{x^2-4x^{8}y(x)}{\tan }^{-1}\left(\sqrt{4x^{6}y(x)-1}\right)}{x\sqrt{4x^{6}y(x)-1}}+\log (y(x))=0,y(x)],\text{Solve}[6c_1+\log (y(x))=\frac{2\sqrt{x^2-4x^{8}y(x)}{\tan }^{-1}\left(\sqrt{4x^{6}y(x)-1}\right)}{x\sqrt{4x^{6}y(x)-1}},y(x)]\}$$

Maple ✓
cpu = 0.069 (sec), leaf count = 71

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-\frac{\ln \left(y\left(x\right)x^6\right)}{6}-\frac{1}{3}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{1-4y\left(x\right)x^6}\right)=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-\frac{\ln \left(y\left(x\right)x^6\right)}{6}+\frac{1}{3}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{1-4y\left(x\right)x^6}\right)=0,y\left(x\right)=\frac{1}{4x^6}\}$$ Mathematica raw input

DSolve[9*y[x] + 3*x*y'[x] + x^8*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[6*C[1] + Log[y[x]] + (2*ArcTan[Sqrt[-1 + 4*x^6*y[x]]]*Sqrt[x^2 - 4*x^8*y[
x]])/(x*Sqrt[-1 + 4*x^6*y[x]]) == 0, y[x]], Solve[6*C[1] + Log[y[x]] == (2*ArcTa
n[Sqrt[-1 + 4*x^6*y[x]]]*Sqrt[x^2 - 4*x^8*y[x]])/(x*Sqrt[-1 + 4*x^6*y[x]]), y[x]
]}

Maple raw input

dsolve(x^8*diff(y(x),x)^2+3*x*diff(y(x),x)+9*y(x) = 0, y(x),'implicit')

Maple raw output

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