68.12.22 problem 22

Internal problem ID [17616]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 22
Date solved : Thursday, October 02, 2025 at 02:26:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+16 y&=\frac {{\mathrm e}^{-4 t}}{t^{4}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+8*diff(y(t),t)+16*y(t) = 1/t^4*exp(-4*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-4 t} \left (c_2 +t c_1 +\frac {1}{6 t^{2}}\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 33

ode=D[y[t],{t,2}]+8*D[y[t],t]+16*y[t]==1/t^4*Exp[-4*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {e^{-4 t} \left (6 c_2 t^3+6 c_1 t^2+1\right )}{6 t^2} \end{align*}

✓ Sympy. Time used: 0.187 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = \left (C_{1} + C_{2} t + \frac {1}{6 t^{2}}\right ) e^{- 4 t} \]