90.8.13 problem 13

Internal problem ID [25187]
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 : 13
Date solved : Thursday, October 02, 2025 at 11:57:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&={\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 14

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = exp(-2*t); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 18

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-2 t} t (t+2) \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 14

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

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