23.4.6 problem 6

Internal problem ID [6308]
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 : 6
Date solved : Tuesday, September 30, 2025 at 02:45:40 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 41.782 (sec). Leaf size: 100

ode=D[y[x],{x,2}] == 2*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 {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\\ y(x)&\to \frac {i \text {sn}\left (\left .(-1)^{3/4} \sqrt {\sqrt {c_1} (x+c_2){}^2}\right |-1\right )}{\sqrt {\frac {i}{\sqrt {c_1}}}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 6.245 (sec). Leaf size: 82

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

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