26.9.9 problem Exercise 22.9, page 240

Internal problem ID [7118]
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 22.9, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

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

\[ y = {\mathrm e}^{-x} \left (c_2 +c_1 x +\frac {x^{2} \left (2 \ln \left (x \right )-3\right )}{4}\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to \frac {1}{4} e^{-x} \left (-3 x^2+2 x^2 \log (x)+4 c_2 x+4 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 22

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)*log(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x \log {\left (x \right )}}{2} - \frac {3 x}{4}\right )\right ) e^{- x} \]