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 : Sunday, March 30, 2025 at 10:30:45 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x) = 2*k*y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

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

ode=D[y[x],{x,2}]==2*k*y[x]^3; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 15.113 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-2*k*y(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {k e^{i \pi } y^{4}{\left (x \right )}}{C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} + x, \ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {k e^{i \pi } y^{4}{\left (x \right )}}{C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} - x\right ] \]