32.10.8 problem Exercise 35.8, page 504

Internal problem ID [6002]
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 35.8, page 504
Date solved : Sunday, March 30, 2025 at 10:30:40 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 15

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

\[ y = \frac {4 \operatorname {WeierstrassP}\left (x +c_1 , 0, c_2\right )}{k} \]

✓ Mathematica. Time used: 1.458 (sec). Leaf size: 36

ode=D[y[x],{x,2}]==3/2*(k*y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {2^{2/3} \wp \left (\frac {\sqrt [3]{k} (x+c_1)}{2^{2/3}};0,c_2\right )}{\sqrt [3]{k}} \]

✓ Sympy. Time used: 15.354 (sec). Leaf size: 85

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

\[ \left [ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {k e^{i \pi } y^{3}{\left (x \right )}}{C_{1}}} \right )}}{3 \sqrt {C_{1}} \Gamma \left (\frac {4}{3}\right )} = C_{2} + x, \ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {k e^{i \pi } y^{3}{\left (x \right )}}{C_{1}}} \right )}}{3 \sqrt {C_{1}} \Gamma \left (\frac {4}{3}\right )} = C_{2} - x\right ] \]