4.18.44 \(8 y(x) y'(x)+16 x y'(x)^2+y(x)^6=0\)

ODE
\[ 8 y(x) y'(x)+16 x y'(x)^2+y(x)^6=0 \] ODE Classification

[[_homogeneous, `class G`]]

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

Mathematica
cpu = 0.262324 (sec), leaf count = 263

\[\left \{\left \{y(x)\to -\frac {\sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to -\frac {i \sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to \frac {i \sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to \frac {\sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to -\frac {\sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to -\frac {i \sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to \frac {i \sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \},\left \{y(x)\to \frac {\sqrt [4]{2} \sqrt [4]{\frac {1}{\cosh \left (c_1-\log (x)\right )+1}}}{\sqrt [4]{x}}\right \}\right \}\]

Maple
cpu = 0.067 (sec), leaf count = 77

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

DSolve[y[x]^6 + 8*y[x]*y'[x] + 16*x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/4)*((1 + Cosh[C[1] - Log[x]])^(-1))^(1/4))/x^(1/4))}, {y[x] ->
 ((-I)*2^(1/4)*((1 + Cosh[C[1] - Log[x]])^(-1))^(1/4))/x^(1/4)}, {y[x] -> (I*2^(
1/4)*((1 + Cosh[C[1] - Log[x]])^(-1))^(1/4))/x^(1/4)}, {y[x] -> (2^(1/4)*((1 + C
osh[C[1] - Log[x]])^(-1))^(1/4))/x^(1/4)}, {y[x] -> -((2^(1/4)*((1 + Cosh[C[1] -
 Log[x]])^(-1))^(1/4))/x^(1/4))}, {y[x] -> ((-I)*2^(1/4)*((1 + Cosh[C[1] - Log[x
]])^(-1))^(1/4))/x^(1/4)}, {y[x] -> (I*2^(1/4)*((1 + Cosh[C[1] - Log[x]])^(-1))^
(1/4))/x^(1/4)}, {y[x] -> (2^(1/4)*((1 + Cosh[C[1] - Log[x]])^(-1))^(1/4))/x^(1/
4)}}

Maple raw input

dsolve(16*x*diff(y(x),x)^2+8*y(x)*diff(y(x),x)+y(x)^6 = 0, y(x),'implicit')

Maple raw output

y(x)^4-1/x = 0, ln(x)-_C1-4*Intat(1/_a/(-_a^4+1)^(1/2),_a = y(x)*x^(1/4)) = 0, l
n(x)-_C1+4*Intat(1/_a/(-_a^4+1)^(1/2),_a = y(x)*x^(1/4)) = 0