74.12.42 problem 42

Internal problem ID [16270]
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, March 13, 2025 at 08:09:00 AM
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: 0.024 (sec). Leaf size: 27

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,ic],y(t), singsol=all);
 

\[ y = \frac {\left (-32 t^{2}-8 t -1\right ) {\mathrm e}^{-4 t}}{32}+\frac {{\mathrm e}^{4 t}}{32} \]

✓ Mathematica. Time used: 0.057 (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]
 

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

✓ Sympy. Time used: 0.165 (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)
 

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