68.11.39 problem 51

Internal problem ID [17576]
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 : 51
Date solved : Thursday, October 02, 2025 at 02:25:38 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+16 y&=4 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {5}{4}} \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)+8*diff(y(t),t)+16*y(t) = 4; 
ic:=[y(0) = 5/4, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-4 t}+4 \,{\mathrm e}^{-4 t} t +\frac {1}{4} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 20

ode=D[y[t],{t,2}]+8*D[y[t],t]+16*y[t]==4; 
ic={y[0]==5/4,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.118 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 4,0) 
ics = {y(0): 5/4, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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