60.3.394 problem 1411

Internal problem ID [11390]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1411
Date solved : Sunday, March 30, 2025 at 08:19:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }&=\frac {y}{{\mathrm e}^{x}+1} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x) = 1/(exp(x)+1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\left (\frac {1}{{\mathrm e}^{x}+1}+\ln \left ({\mathrm e}^{x}+1\right )\right ) c_1 +c_2 \right ) \left (1+{\mathrm e}^{-x}\right ) \]

✓ Mathematica. Time used: 0.664 (sec). Leaf size: 81

ode=D[y[x],{x,2}] == y[x]/(1 + E^x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\exp \left (\int _1^{e^x}\left (\frac {1}{K[1]+1}-\frac {1}{2 K[1]}\right )dK[1]\right ) \left (c_2 \int _1^{e^x}\exp \left (-2 \int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{2 K[1]}\right )dK[1]\right )dK[2]+c_1\right )}{\sqrt {e^x}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - y(x)/(exp(x) + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False