ODE
\[ -4 x^2 y(x) y'(x)^2+y'(x)^4+16 x y(x)^2 y'(x)-16 y(x)^3=0 \] ODE Classification
[[_homogeneous, `class G`]]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 1.73458 (sec), leaf count = 341
\[\left \{\text {Solve}\left [\frac {1}{4} \left (-4 c_1+\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)} \left (4 \log \left (\sqrt {x^2-4 \sqrt {y(x)}}+x\right )-\log (y(x))\right )}{\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )=0,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (-4 c_1+\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)} \left (\log (y(x))-4 \log \left (\sqrt {x^2-4 \sqrt {y(x)}}+x\right )\right )}{\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )=0,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (-4 c_1+\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \left (4 \log \left (\sqrt {x^2+4 \sqrt {y(x)}}+x\right )-\log (y(x))\right )}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )=0,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (-4 c_1+\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \left (\log (y(x))-4 \log \left (\sqrt {x^2+4 \sqrt {y(x)}}+x\right )\right )}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )=0,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.311 (sec), leaf count = 118
\[ \left \{ {1 \left ( \sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}-x \right ) ^{{1\sqrt {{x}^{2}y \left ( x \right ) -4\, \left ( y \left ( x \right ) \right ) ^{3/2}}{\frac {1}{\sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}}}{\frac {1}{\sqrt {y \left ( x \right ) }}}}}{\frac {1}{\sqrt {y \left ( x \right ) }}} \left ( \left ( \sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}+x \right ) ^{{1\sqrt {{x}^{2}y \left ( x \right ) -4\, \left ( y \left ( x \right ) \right ) ^{3/2}}{\frac {1}{\sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}}}{\frac {1}{\sqrt {y \left ( x \right ) }}}}} \right ) ^{-1}}-{\it \_C1}=0,y \left ( x \right ) ={\frac {{x}^{4}}{16}} \right \} \] Mathematica raw input
DSolve[-16*y[x]^3 + 16*x*y[x]^2*y'[x] - 4*x^2*y[x]*y'[x]^2 + y'[x]^4 == 0,y[x],x]
Mathematica raw output
{Solve[(-4*C[1] + Log[y[x]] + ((4*Log[x + Sqrt[x^2 - 4*Sqrt[y[x]]]] - Log[y[x]])*Sqrt[x^2 - 4*Sqrt[y[x]]]*Sqrt[y[x]])/Sqrt[(x^2 - 4*Sqrt[y[x]])*y[x]])/4 == 0, y[x]], Solve[(-4*C[1] + Log[y[x]] + ((-4*Log[x + Sqrt[x^2 - 4*Sqrt[y[x]]]] + Log[y[x]])*Sqrt[x^2 - 4*Sqrt[y[x]]]*Sqrt[y[x]])/Sqrt[(x^2 - 4*Sqrt[y[x]])*y[x]])/4 == 0, y[x]], Solve[(-4*C[1] + Log[y[x]] + ((4*Log[x + Sqrt[x^2 + 4*Sqrt[y[x]]]] - Log[y[x]])*Sqrt[x^2 + 4*Sqrt[y[x]]]*Sqrt[y[x]])/Sqrt[(x^2 + 4*Sqrt[y[x]])*y[x]])/4 == 0, y[x]], Solve[(-4*C[1] + Log[y[x]] + ((-4*Log[x + Sqrt[x^2 + 4*Sqrt[y[x]]]] + Log[y[x]])*Sqrt[x^2 + 4*Sqrt[y[x]]]*Sqrt[y[x]])/Sqrt[(x^2 + 4*Sqrt[y[x]])*y[x]])/4 == 0, y[x]]}
Maple raw input
dsolve(diff(y(x),x)^4-4*x^2*y(x)*diff(y(x),x)^2+16*x*y(x)^2*diff(y(x),x)-16*y(x)^3 = 0, y(x),'implicit')
Maple raw output
y(x) = 1/16*x^4, 1/y(x)^(1/2)/(((x^2-4*y(x)^(1/2))^(1/2)+x)^(1/(x^2-4*y(x)^(1/2))^(1/2)/y(x)^(1/2)*(x^2*y(x)-4*y(x)^(3/2))^(1/2)))*((x^2-4*y(x)^(1/2))^(1/2)-x)^(1/(x^2-4*y(x)^(1/2))^(1/2)/y(x)^(1/2)*(x^2*y(x)-4*y(x)^(3/2))^(1/2))-_C1 = 0