70.18.3 problem 14

Internal problem ID [19000]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 14
Date solved : Thursday, October 02, 2025 at 03:36:52 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = 0; 
ic:=[y(0) = 3, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -2 \,{\mathrm e}^{-2 t}+5 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 18

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==0; 
ic={y[0]==3,Derivative[1][y][0] == -1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (5 e^t-2\right ) \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (5 - 2 e^{- t}\right ) e^{- t} \]