89.25.1 problem 1

Internal problem ID [24859]
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 : 1
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 ({\mathrm e}^{x}-1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.653 (sec). Leaf size: 53

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^{-x} \left (\exp ^3 (c_2 x+c_1)+e^{\frac {1}{\exp }} (\exp (x-1)-1) \operatorname {ExpIntegralEi}\left (x-\frac {1}{\exp }\right )-\exp e^x\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