69.20.15 problem 654

Internal problem ID [18436]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 654
Date solved : Thursday, October 02, 2025 at 03:11:53 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 40

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

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

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to x-\log \left (e^x+1\right )-e^{-x} \left (\log \left (e^x+1\right )+c_1\right )+c_2 \end{align*}

✗ Sympy

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

Timed Out