4.19.42 $4x^5{y}^{\prime }(x{)}^2+12x^{4}y(x)y^{\prime }(x)+9=0$

ODE
$$4x^5{y}^{\prime }(x{)}^2+12x^{4}y(x)y^{\prime }(x)+9=0$$ ODE Classification

[[_homogeneous, `class G`]]

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

Mathematica ✓
cpu = 0.706901 (sec), leaf count = 121

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

Maple ✓
cpu = 0.158 (sec), leaf count = 67

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

DSolve[9 + 12*x^4*y[x]*y'[x] + 4*x^5*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[Log[x] == C[1] + (2*x^(5/2)*ArcSin[x^(3/2)*y[x]]*Sqrt[1 - x^3*y[x]^2])/(3
*Sqrt[x^5*(-1 + x^3*y[x]^2)]), y[x]], Solve[Log[x] + (2*x^(5/2)*ArcSin[x^(3/2)*y
[x]]*Sqrt[1 - x^3*y[x]^2])/(3*Sqrt[x^5*(-1 + x^3*y[x]^2)]) == C[1], y[x]]}

Maple raw input

dsolve(4*x^5*diff(y(x),x)^2+12*x^4*y(x)*diff(y(x),x)+9 = 0, y(x),'implicit')

Maple raw output

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