44.11.3 problem 1(c)

Internal problem ID [9273]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 1(c)
Date solved : Tuesday, September 30, 2025 at 06:15:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=64 x \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = 64*x*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 50

ode=D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]==64*x*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (e^{4 x} \int _1^x16 e^{-4 K[1]} K[1]dK[1]-8 x^2+c_2 e^{4 x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-64*x*exp(-x) - 3*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} e^{3 x} + \left (C_{1} - 8 x^{2} - 4 x\right ) e^{- x} \]