4.18.42 \(4 y(x) y'(x)+4 x y'(x)^2-y(x)^4=0\)

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

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for \(x\)

Mathematica
cpu = 0.194342 (sec), leaf count = 123

\[\left \{\left \{y(x)\to -\frac {\sqrt {\tanh ^2\left (\frac {1}{2} \left (c_1-\log (x)\right )\right )-1}}{\sqrt {x}}\right \},\left \{y(x)\to \frac {\sqrt {\tanh ^2\left (\frac {1}{2} \left (c_1-\log (x)\right )\right )-1}}{\sqrt {x}}\right \},\left \{y(x)\to -\frac {\sqrt {\tanh ^2\left (\frac {1}{2} \left (\log (x)-c_1\right )\right )-1}}{\sqrt {x}}\right \},\left \{y(x)\to \frac {\sqrt {\tanh ^2\left (\frac {1}{2} \left (\log (x)-c_1\right )\right )-1}}{\sqrt {x}}\right \}\right \}\]

Maple
cpu = 0.073 (sec), leaf count = 53

\[ \left \{ \left (y \relax (x ) \right ) ^{2}+{x}^{-1}=0,\ln \relax (x ) -{\it \_C1}-2\,{\it Artanh} \left ({\frac {1}{\sqrt {1+x \left (y \relax (x ) \right ) ^{2}}}} \right ) =0,\ln \relax (x ) -{\it \_C1}+2\,{\it Artanh} \left ({\frac {1}{\sqrt {1+x \left (y \relax (x ) \right ) ^{2}}}} \right ) =0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[-1 + Tanh[(C[1] - Log[x])/2]^2]/Sqrt[x])}, {y[x] -> Sqrt[-1 + T
anh[(C[1] - Log[x])/2]^2]/Sqrt[x]}, {y[x] -> -(Sqrt[-1 + Tanh[(-C[1] + Log[x])/2
]^2]/Sqrt[x])}, {y[x] -> Sqrt[-1 + Tanh[(-C[1] + Log[x])/2]^2]/Sqrt[x]}}

Maple raw input

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

Maple raw output

y(x)^2+1/x = 0, ln(x)-_C1+2*arctanh(1/(1+x*y(x)^2)^(1/2)) = 0, ln(x)-_C1-2*arcta
nh(1/(1+x*y(x)^2)^(1/2)) = 0