4.17.10 \(4 x^5 y'(x)-12 x^4 y(x)+y'(x)^2=0\)

ODE
\[ 4 x^5 y'(x)-12 x^4 y(x)+y'(x)^2=0 \] 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

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

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

\[ \left \{ y \relax (x ) =-{\frac {{x}^{6}}{3}},y \relax (x ) ={\it \_C1}\,{x}^{3}+{\frac {3\,{{\it \_C1}}^{2}}{4}} \right \} \] 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