89.19.18 problem 18

Internal problem ID [24746]
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 : 18
Date solved : Thursday, October 02, 2025 at 10:47:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=16 x \,{\mathrm e}^{-2 x}+8 x +4 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)-4*y(x) = 16*x*exp(-2*x)+8*x+4; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-8 x^{2}+4 c_1 -4 x -1\right ) {\mathrm e}^{-2 x}}{4}+{\mathrm e}^{2 x} c_2 -2 x -1 \]

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 49

ode=D[y[x],{x,2}]-4*y[x]==16*x*Exp[-2*x]+8*x+4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^{-2 x} \left (-8 x^2-4 x-4 e^{2 x} (2 x+1)+4 c_1 e^{4 x}-1+4 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x - 16*x*exp(-2*x) - 4*y(x) + Derivative(y(x), (x, 2)) - 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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