23.4.3 problem 3

Internal problem ID [6305]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 02:45:37 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=6 y^{2} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 10

ode:=diff(diff(y(x),x),x) = 6*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 14

ode=D[y[x],{x,2}] == 6*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \wp (x+c_1;0,c_2) \end{align*}

✓ Sympy. Time used: 6.207 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*y(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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