4.18.44 $8y(x)y^{\prime }(x)+16x{y}^{\prime }(x{)}^2+y(x{)}^6=0$

ODE
$$8y(x)y^{\prime }(x)+16x{y}^{\prime }(x{)}^2+y(x{)}^6=0$$ ODE Classification

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for $x$

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

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

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

$$\{{\left(y\left(x\right)\right)}^4-x^{-1}=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-4{\int }^{y\left(x\right)\sqrt[4]{x}}\frac{1}{\mathit{\_}\mathit{a}\sqrt{-{\mathit{\_}\mathit{a}}^4+1}}d\mathit{\_}\mathit{a}=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+4{\int }^{y\left(x\right)\sqrt[4]{x}}\frac{1}{\mathit{\_}\mathit{a}\sqrt{-{\mathit{\_}\mathit{a}}^4+1}}d\mathit{\_}\mathit{a}=0\}$$ 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