54.7.24 problem 1614 (6.24)

Internal problem ID [12873]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1614 (6.24)
Date solved : Friday, October 03, 2025 at 03:49:42 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+3*a*diff(y(x),x)-2*y(x)^3+2*a^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.688 (sec). Leaf size: 32

ode=2*a^2*y[x] - 2*y[x]^3 + 3*a*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*a**2*y(x) + 2*y(x)**3 - Derivative(y(x), (x, 2)))/(3*a) cannot be solved by the factorable group method