Internal problem ID [9176]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1597.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 21
dsolve(diff(diff(y(x),x),x)-a*y(x)^3=0,y(x), singsol=all)
\[ y \relax (x ) = c_{2} \mathrm {sn}\left (\left (\frac {\sqrt {-2 a}\, x}{2}+c_{1}\right ) c_{2}| i\right ) \]
✓ Solution by Mathematica
Time used: 1.275 (sec). Leaf size: 131
DSolve[-(a*y[x]^3) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\ y(x)\to \frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\ \end{align*}