Internal problem ID [4151]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 8. Special second order equations. Lesson 35. Independent variable x
absent
Problem number: Exercise 35.9, page 504.
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 }-2 k y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 20
dsolve(diff(y(x),x$2)=2*k*y(x)^3,y(x), singsol=all)
\[ y \relax (x ) = c_{2} \mathrm {sn}\left (\left (\sqrt {-k}\, x +c_{1}\right ) c_{2}| i\right ) \]
✓ Solution by Mathematica
Time used: 1.095 (sec). Leaf size: 115
DSolve[y''[x]==2*k*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {k} \sqrt {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i \sqrt {k}}{\sqrt {c_1}}}} \\ y(x)\to \frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {k} \sqrt {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i \sqrt {k}}{\sqrt {c_1}}}} \\ \end{align*}