90.8.11 problem 11

Internal problem ID [25185]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 109
Problem number : 11
Date solved : Thursday, October 02, 2025 at 11:57:38 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+16 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 8

ode:=diff(diff(y(t),t),t)+8*diff(y(t),t)+16*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = -4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 10

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

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

✓ Sympy. Time used: 0.093 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -4} 
dsolve(ode,func=y(t),ics=ics)
 

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