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 = 1.18291 (sec), leaf count = 359
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 i \text {$\#$1} \sqrt {\frac {c_1}{\text {$\#$1} \left (2-2 \sqrt {1-c_1}\right )}+1} \sqrt {\frac {c_1}{\text {$\#$1} \left (2 \sqrt {1-c_1}+2\right )}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 i \text {$\#$1} \sqrt {\frac {c_1}{\text {$\#$1} \left (2-2 \sqrt {1-c_1}\right )}+1} \sqrt {\frac {c_1}{\text {$\#$1} \left (2 \sqrt {1-c_1}+2\right )}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.075 (sec), leaf count = 61
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {4\,{{\it \_a}}^{3}+{\it \_a}\,{\it \_C1}+4\,{{\it \_a}}^{2}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt { \left ( 4\,{{\it \_a}}^{2}+{\it \_C1}+4\,{\it \_a} \right ) {\it \_a}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] 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[((-2*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]])]*Sqrt[1 + C[1]/((2 - 2*Sqrt[1 - C[1]])*#1)]*Sqrt[1 + C[1]/((2 + 2*Sqrt[1 - C[1]])*#1)]*#1)/(Sqrt[C[1]/(2 + 2*Sqrt[1 - C[1]])]*Sqrt[C[1] + 4*#1 + 4*#1^2]) & ][x + C[2]]}, {y[x] -> InverseFunction[((2*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]])]*Sqrt[1 + C[1]/((2 - 2*Sqrt[1 - C[1]])*#1)]*Sqrt[1 + C[1]/((2 + 2*Sqrt[1 - C[1]])*#1)]*#1)/(Sqrt[C[1]/(2 + 2*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),'implicit')
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^2+_C1+4*_a)*_a)^(1/2),_a = y(x))-x-_C2 = 0