4.17.10 $4x^5{y}^{\prime }(x)-12x^{4}y(x)+y^{\prime }(x{)}^2=0$

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

[[_1st_order, _with_linear_symmetries]]

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

Mathematica ✓
cpu = 1.09311 (sec), leaf count = 217

$$\{\text{Solve}[\frac{x^2\sqrt{x^6+3y(x)}\log (y(x))+\sqrt{x^4(x^6+3y(x))}(\log (\frac{x^6}{3y(x)}+1)-\log (\frac{3y(x)}{x^6}+1)+2\log (\sqrt{x^6+3y(x)}+x^3))}{6x^2\sqrt{x^6+3y(x)}}=c_1,y(x)],\text{Solve}[\frac{x^2\sqrt{x^6+3y(x)}\log (y(x))+\sqrt{x^4(x^6+3y(x))}(-\log (\frac{x^6}{3y(x)}+1)+\log (\frac{3y(x)}{x^6}+1)-2\log (\sqrt{x^6+3y(x)}+x^3))}{6x^2\sqrt{x^6+3y(x)}}=c_1,y(x)]\}$$

Maple ✓
cpu = 0.205 (sec), leaf count = 23

$$\{y\left(x\right)=-\frac{x^6}{3},y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{x}^3+\frac{3{\mathit{\_}\mathit{C}\mathit{1}}^2}{4}\}$$ Mathematica raw input

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

Mathematica raw output

{Solve[(x^2*Log[y[x]]*Sqrt[x^6 + 3*y[x]] + (Log[1 + x^6/(3*y[x])] - Log[1 + (3*y
[x])/x^6] + 2*Log[x^3 + Sqrt[x^6 + 3*y[x]]])*Sqrt[x^4*(x^6 + 3*y[x])])/(6*x^2*Sq
rt[x^6 + 3*y[x]]) == C[1], y[x]], Solve[(x^2*Log[y[x]]*Sqrt[x^6 + 3*y[x]] + (-Lo
g[1 + x^6/(3*y[x])] + Log[1 + (3*y[x])/x^6] - 2*Log[x^3 + Sqrt[x^6 + 3*y[x]]])*S
qrt[x^4*(x^6 + 3*y[x])])/(6*x^2*Sqrt[x^6 + 3*y[x]]) == C[1], y[x]]}

Maple raw input

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

Maple raw output

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