89.23.8 problem 8

Internal problem ID [24812]
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. Miscellaneous Exercises at page 162
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:48:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=4 x^{2}-3 \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)-diff(y(x),x)-2*y(x) = 4*x^2-3*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 46

ode=D[y[x],{x,2}]-D[y[x],{x,1}]-2*y[x]==4*x^2-3*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{-x} \left (e^x \left (-6 x^2+6 x-9\right )+3 x+3 c_2 e^{3 x}+1+3 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2 - 2*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 3*exp(-x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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