32.10.9 problem Exercise 35.9, page 504

Internal problem ID [6003]
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
Date solved : Monday, January 27, 2025 at 01:31:59 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=2 k y^{3} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 19

dsolve(diff(y(x),x$2)=2*k*y(x)^3,y(x), singsol=all)
 

\[ y = c_2 \,\operatorname {JacobiSN}\left (\left (\sqrt {-k}\, x +c_1 \right ) c_2 , i\right ) \]

✓ Solution by Mathematica

Time used: 61.182 (sec). Leaf size: 115

DSolve[D[y[x],{x,2}]==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*}