68.12.5 problem 5

Internal problem ID [17599]
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 : 5
Date solved : Thursday, October 02, 2025 at 02:26:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+20 y&=2 t \,{\mathrm e}^{-2 t} \end{align*}

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

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+20*y(t) = 2*t*exp(-2*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 32

ode=D[y[t],{t,2}]+4*D[y[t],t]+20*y[t]==2*t*Exp[-2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} e^{-2 t} (t+8 c_2 \cos (4 t)+8 c_1 \sin (4 t)) \end{align*}

✓ Sympy. Time used: 0.166 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*exp(-2*t) + 20*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )} + \frac {t}{8}\right ) e^{- 2 t} \]