89.19.4 problem 4

Internal problem ID [24732]
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 : 4
Date solved : Thursday, October 02, 2025 at 10:47:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y+2 y^{\prime }+y^{\prime \prime }&=3 x \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 3*x*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} \left (c_2 +c_1 x +\frac {1}{2} x^{3}\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+2*D[y[x],{x,1}]+y[x]==3*x*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.154 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*exp(-x) + y(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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