89.19.6 problem 6

Internal problem ID [24734]
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 : 6
Date solved : Thursday, October 02, 2025 at 10:47:32 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+12 y^{\prime \prime }+48 y^{\prime }+64 y&=8 x \,{\mathrm e}^{-4 x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

ode:=diff(diff(diff(y(x),x),x),x)+12*diff(diff(y(x),x),x)+48*diff(y(x),x)+64*y(x) = 8*x*exp(-4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-4 x} \left (\frac {1}{3} x^{4}+c_1 +c_2 x +c_3 \,x^{2}\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 34

ode=D[y[x],{x,3}]+12*D[y[x],{x,2}]+48*D[y[x],{x,1}]+64*y[x]==8*x*Exp[-4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{-4 x} \left (x^4+3 c_3 x^2+3 c_2 x+3 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.237 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x*exp(-4*x) + 64*y(x) + 48*Derivative(y(x), x) + 12*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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