61.3.3 problem Problem 4

Internal problem ID [15302]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 4
Date solved : Thursday, October 02, 2025 at 10:11:28 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.095 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 14

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

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

✓ Sympy. Time used: 0.091 (sec). Leaf size: 8

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

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