89.19.3 problem 3

Internal problem ID [24731]
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 : 3
Date solved : Thursday, October 02, 2025 at 10:47:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=12 x \,{\mathrm e}^{-2 x} \end{align*}

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

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 12*x*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 23

ode=D[y[x],{x,2}]+4*D[y[x],{x,1}]+4*y[x]==12*x*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.170 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*x*exp(-2*x) + 4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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