ODE
\[ 2 y(x) y''(x)=y'(x)^2+8 y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.01736 (sec), leaf count = 135
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.325 (sec), leaf count = 53
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}+\textit {\_a} \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+\textit {\_a} \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[2*y[x]*y''[x] == 8*y[x]^3 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-2*Hypergeometric2F1[1/4, 1/2, 5/4, (-4*#1^2)/C[1]]*Sqrt[#1]*Sqrt[1 + (4*#1^2)/C[1]])/Sqrt[C[1] + 4*#1^2] & ][x + C[2]]}, {y[x] -> InverseFunction[(2*Hypergeometric2F1[1/4, 1/2, 5/4, (-4*#1^2)/C[1]]*Sqrt[#1]*Sqrt[1 + (4*#1^2)/C[1]])/Sqrt[C[1] + 4*#1^2] & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = 8*y(x)^3+diff(y(x),x)^2, y(x))
Maple raw output
[Intat(1/(4*_a^3+_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(4*_a^3+_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0]