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 = 0.427528 (sec), leaf count = 95
\[\left \{\left \{y(x)\to -\frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\right \},\left \{y(x)\to \frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\right \}\right \}\]
Maple ✓
cpu = 0.052 (sec), leaf count = 17
\[[y \left (x \right ) = \textit {\_C2} \,\mathrm {sn}\left (\left (i x +\textit {\_C1} \right ) \textit {\_C2} | i\right )]\] 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))
Maple raw output
[y(x) = _C2*JacobiSN((I*x+_C1)*_C2,I)]