23.4.74 problem 74

Internal problem ID [6376]
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 : 74
Date solved : Tuesday, September 30, 2025 at 02:54:48 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 2 y^{\prime \prime }&=y \left (a -y^{2}\right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 65

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

\[ y = c_2 \sqrt {2}\, \sqrt {\frac {a}{c_2^{2}+2 a -1}}\, \operatorname {JacobiSN}\left (\frac {\left (\sqrt {-2 a +1}\, x +2 c_1 \right ) \sqrt {2}\, \sqrt {\frac {a}{c_2^{2}+2 a -1}}}{2}, \frac {c_2}{\sqrt {2 a -1}}\right ) \]

✓ Mathematica. Time used: 60.104 (sec). Leaf size: 189

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

\begin{align*} y(x)&\to -\frac {i \text {sn}\left (\frac {1}{2} \sqrt {-\left (\left (a+\sqrt {a^2+4 c_1}\right ) (x+c_2){}^2\right )}|\frac {a-\sqrt {a^2+4 c_1}}{a+\sqrt {a^2+4 c_1}}\right )}{\sqrt {\frac {1}{-a+\sqrt {a^2+4 c_1}}}}\\ y(x)&\to \frac {i \text {sn}\left (\frac {1}{2} \sqrt {-\left (\left (a+\sqrt {a^2+4 c_1}\right ) (x+c_2){}^2\right )}|\frac {a-\sqrt {a^2+4 c_1}}{a+\sqrt {a^2+4 c_1}}\right )}{\sqrt {\frac {1}{-a+\sqrt {a^2+4 c_1}}}} \end{align*}

✗ Sympy

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

Timed Out