68.11.31 problem 43

Internal problem ID [17568]
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 : 43
Date solved : Thursday, October 02, 2025 at 02:25:31 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }&=-24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t) = -24*t-6-4*t*exp(-4*t)+exp(-4*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 2.712 (sec). Leaf size: 75

ode=D[y[t],{t,2}]+4*D[y[t],t]==-24*t-6-4*t*Exp[-4*t]+Exp[-4*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \int _1^te^{-4 K[1]} \left (c_1-K[1] \left (2 K[1]+6 e^{4 K[1]}-1\right )\right )dK[1]+c_2\\ y(t)&\to \frac {1}{2} e^{-4 t} t^2-3 t^2-\frac {1}{2 e^4}+3+c_2 \end{align*}

✓ Sympy. Time used: 0.188 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(24*t + 4*t*exp(-4*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 6 - exp(-4*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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