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

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

[[_homogeneous, `class G`]]

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

Mathematica
cpu = 0.204213 (sec), leaf count = 133

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

Maple
cpu = 0.072 (sec), leaf count = 57

\[ \left \{ \left (y \relax (x ) \right ) ^{2}+{\frac {1}{4\,x}}=0,\ln \relax (x ) -{\it \_C1}-2\,{\it Artanh} \left ({\frac {1}{\sqrt {4\,x \left (y \relax (x ) \right ) ^{2}+1}}} \right ) =0,\ln \relax (x ) -{\it \_C1}+2\,{\it Artanh} \left ({\frac {1}{\sqrt {4\,x \left (y \relax (x ) \right ) ^{2}+1}}} \right ) =0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x)^2+1/4/x = 0, ln(x)-_C1+2*arctanh(1/(4*x*y(x)^2+1)^(1/2)) = 0, ln(x)-_C1-2*a
rctanh(1/(4*x*y(x)^2+1)^(1/2)) = 0