89.25.2 problem 2

Internal problem ID [24860]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 177
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:48:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y+2 y^{\prime }+y^{\prime \prime }&=\frac {1}{\left (1+{\mathrm e}^{x}\right )^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 1/(exp(x)+1)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} \left (c_2 +c_1 x +\ln \left ({\mathrm e}^{x}+1\right )-\frac {x}{2}\right ) \]

✓ Mathematica. Time used: 0.682 (sec). Leaf size: 59

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

\begin{align*} y(x)&\to \frac {e^{-\frac {1}{\exp }-x} \left ((\exp (x-1)+1) \operatorname {ExpIntegralEi}\left (x+\frac {1}{\exp }\right )+e^{\frac {1}{\exp }} \exp \left (-e^x+\exp ^2 (c_2 x+c_1)\right )\right )}{\exp ^3} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-y(x)*exp(2*x) - 2*y(x)*exp(x) - y(x) - e