68.12.42 problem 42

Internal problem ID [17636]
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 : 42
Date solved : Thursday, October 02, 2025 at 02:26:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-16*y(t) = 16*t*exp(-4*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\left (-4 t^{2}-t \right ) {\mathrm e}^{-4 t}}{4}+\frac {\sinh \left (4 t \right )}{16} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 63

ode=D[y[t],{t,2}]-16*y[t]==16*t*Exp[-4*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{-4 t} \left (e^{8 t} \int _1^02 e^{-8 K[1]} K[1]dK[1]-e^{8 t} \int _1^t2 e^{-8 K[1]} K[1]dK[1]+t^2\right ) \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \left (- t^{2} - \frac {t}{4} - \frac {1}{32}\right ) e^{- 4 t} + \frac {e^{4 t}}{32} \]