26.9.12 problem Exercise 22.12, page 240

Internal problem ID [7121]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22.12, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=\frac {{\mathrm e}^{-x}}{x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 0.159 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \log {\left (x \right )}\right )\right ) e^{- x} \]