7.19.4 problem 30

Internal problem ID [544]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 30
Date solved : Tuesday, March 04, 2025 at 11:26:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+8 x&={\mathrm e}^{-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: 0.232 (sec). Leaf size: 30

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+8*x(t) = exp(-t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = \frac {\left (-2 \cos \left (2 t \right )-\sin \left (2 t \right )\right ) {\mathrm e}^{-2 t}}{10}+\frac {{\mathrm e}^{-t}}{5} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 32

ode=D[x[t],{t,2}]+4*D[x[t],t]+8*x[t]==Exp[-t]; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{10} e^{-2 t} \left (2 e^t-\sin (2 t)-2 \cos (2 t)\right ) \]

✓ Sympy. Time used: 0.278 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(8*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(-t),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\left (- \frac {\sin {\left (2 t \right )}}{10} - \frac {\cos {\left (2 t \right )}}{5}\right ) e^{- t} + \frac {1}{5}\right ) e^{- t} \]