57.16.2 problem 8

Internal problem ID [14480]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.3 The convolution property. Exercises page 162
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:37:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+3 x^{\prime }+2 x&={\mathrm e}^{-4 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.089 (sec). Leaf size: 22

ode:=diff(diff(x(t),t),t)+3*diff(x(t),t)+2*x(t) = exp(-4*t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {\left (2 \,{\mathrm e}^{t}+1\right ) \left ({\mathrm e}^{t}-1\right )^{2} {\mathrm e}^{-4 t}}{6} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 28

ode=D[x[t],{t,2}]+3*D[x[t],t]+2*x[t]==Exp[-4*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{-4 t} \left (e^t-1\right )^2 \left (2 e^t+1\right ) \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*x(t) + 3*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(-4*t),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {1}{3} - \frac {e^{- t}}{2} + \frac {e^{- 3 t}}{6}\right ) e^{- t} \]