ODE
\[ 2 y(x) y''(x)=y'(x)^2+8 y(x)^3+4 y(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.24978 (sec), leaf count = 351
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {i \text {$\#$1} \sqrt {4+\frac {2 c_1}{\text {$\#$1}-\text {$\#$1} \sqrt {1-c_1}}} \sqrt {2+\frac {c_1}{\text {$\#$1}+\text {$\#$1} \sqrt {1-c_1}}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{1+\sqrt {1-c_1}}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {i \text {$\#$1} \sqrt {4+\frac {2 c_1}{\text {$\#$1}-\text {$\#$1} \sqrt {1-c_1}}} \sqrt {2+\frac {c_1}{\text {$\#$1}+\text {$\#$1} \sqrt {1-c_1}}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{1+\sqrt {1-c_1}}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.335 (sec), leaf count = 63
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}+\textit {\_a} \textit {\_C1} +4 \textit {\_a}^{2}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+\textit {\_a} \textit {\_C1} +4 \textit {\_a}^{2}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[2*y[x]*y''[x] == 4*y[x]^2 + 8*y[x]^3 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[((-I)*EllipticF[I*ArcSinh[Sqrt[C[1]/(2 + 2*Sqrt[1 - C[1]])]/Sqrt[#1]], (1 + Sqrt[1 - C[1]])/(1 - Sqrt[1 - C[1]])]*#1*Sqrt[4 + (2*C[1])/(#1 - Sqrt[1 - C[1]]*#1)]*Sqrt[2 + C[1]/(#1 + Sqrt[1 - C[1]]*#1)])/(Sqrt[C[1]/(1 + Sqrt[1 - C[1]])]*Sqrt[C[1] + 4*#1 + 4*#1^2]) & ][x + C[2]]}, {y[x] -> InverseFunction[(I*EllipticF[I*ArcSinh[Sqrt[C[1]/(2 + 2*Sqrt[1 - C[1]])]/Sqrt[#1]], (1 + Sqrt[1 - C[1]])/(1 - Sqrt[1 - C[1]])]*#1*Sqrt[4 + (2*C[1])/(#1 - Sqrt[1 - C[1]]*#1)]*Sqrt[2 + C[1]/(#1 + Sqrt[1 - C[1]]*#1)])/(Sqrt[C[1]/(1 + Sqrt[1 - C[1]])]*Sqrt[C[1] + 4*#1 + 4*#1^2]) & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+4*y(x)^2+8*y(x)^3, y(x))
Maple raw output
[Intat(1/(4*_a^3+_C1*_a+4*_a^2)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(4*_a^3+_C1*_a+4*_a^2)^(1/2),_a = y(x))-x-_C2 = 0]