68.11.54 problem 62 (e)

Internal problem ID [17591]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 62 (e)
Date solved : Thursday, October 02, 2025 at 02:25:58 PM
CAS classification : [[_linear, `class A`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.099 (sec). Leaf size: 14

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

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