ODE
\[ y''(x)=2 y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.338771 (sec), leaf count = 95
\[\left \{\left \{y(x)\to -\frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {c_1} \left (x+c_2\right ){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\right \},\left \{y(x)\to \frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {c_1} \left (x+c_2\right ){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\right \}\right \}\]
Maple ✓
cpu = 0.027 (sec), leaf count = 17
\[ \left \{ y \left ( x \right ) ={\it \_C2}\,{\it JacobiSN} \left ( \left ( ix+{\it \_C1} \right ) {\it \_C2},i \right ) \right \} \] Mathematica raw input
DSolve[y''[x] == 2*y[x]^3,y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*JacobiSN[(-1)^(3/4)*Sqrt[Sqrt[C[1]]*(x + C[2])^2], -1])/Sqrt[I/Sqrt[C[1]]]}, {y[x] -> (I*JacobiSN[(-1)^(3/4)*Sqrt[Sqrt[C[1]]*(x + C[2])^2], -1])/Sqrt[I/Sqrt[C[1]]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = 2*y(x)^3, y(x),'implicit')
Maple raw output
y(x) = _C2*JacobiSN((I*x+_C1)*_C2,I)