89.19.26 problem 26

Internal problem ID [24754]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 26
Date solved : Thursday, October 02, 2025 at 10:47:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=-\frac {{\mathrm e}^{-2 x}}{x^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = -1/x^2*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 21

ode=D[y[x],{x,2}]+4*D[y[x],{x,1}]+4*y[x]== -1/x^2*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.183 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + exp(-2*x)/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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