ODE
\[ 2 y''(x)=y(x) \left (a-y(x)^2\right ) \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.391399 (sec), leaf count = 189
\[\left \{\left \{y(x)\to -\frac {i \text {sn}\left (\frac {1}{2} \sqrt {-\left (a+\sqrt {a^2+4 c_1}\right ) \left (x+c_2\right ){}^2}|\frac {a-\sqrt {a^2+4 c_1}}{a+\sqrt {a^2+4 c_1}}\right )}{\sqrt {\frac {1}{\sqrt {a^2+4 c_1}-a}}}\right \},\left \{y(x)\to \frac {i \text {sn}\left (\frac {1}{2} \sqrt {-\left (a+\sqrt {a^2+4 c_1}\right ) \left (x+c_2\right ){}^2}|\frac {a-\sqrt {a^2+4 c_1}}{a+\sqrt {a^2+4 c_1}}\right )}{\sqrt {\frac {1}{\sqrt {a^2+4 c_1}-a}}}\right \}\right \}\]
Maple ✓
cpu = 0.221 (sec), leaf count = 63
\[ \left \{ y \left ( x \right ) ={\it \_C2}\,\sqrt {2}\sqrt {{\frac {a}{{{\it \_C2}}^{2}+2\,a-1}}}{\it JacobiSN} \left ( \left ( {\frac {x}{2}\sqrt {1-2\,a}}+{\it \_C1} \right ) \sqrt {2}\sqrt {{\frac {a}{{{\it \_C2}}^{2}+2\,a-1}}},{{\it \_C2}{\frac {1}{\sqrt {2\,a-1}}}} \right ) \right \} \] Mathematica raw input
DSolve[2*y''[x] == y[x]*(a - y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*JacobiSN[Sqrt[-((a + Sqrt[a^2 + 4*C[1]])*(x + C[2])^2)]/2, (a - Sqrt[a^2 + 4*C[1]])/(a + Sqrt[a^2 + 4*C[1]])])/Sqrt[(-a + Sqrt[a^2 + 4*C[1]])^(-1)]}, {y[x] -> (I*JacobiSN[Sqrt[-((a + Sqrt[a^2 + 4*C[1]])*(x + C[2])^2)]/2, (a - Sqrt[a^2 + 4*C[1]])/(a + Sqrt[a^2 + 4*C[1]])])/Sqrt[(-a + Sqrt[a^2 + 4*C[1]])^(-1)]}}
Maple raw input
dsolve(2*diff(diff(y(x),x),x) = y(x)*(a-y(x)^2), y(x),'implicit')
Maple raw output
y(x) = _C2*2^(1/2)*(a/(_C2^2+2*a-1))^(1/2)*JacobiSN((1/2*(1-2*a)^(1/2)*x+_C1)*2^(1/2)*(a/(_C2^2+2*a-1))^(1/2),_C2/(2*a-1)^(1/2))