58.20.8 problem 8

Internal problem ID [14932]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:56:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.169 (sec). Leaf size: 20

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

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