63.15.9 problem 6(i)

Internal problem ID [13113]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 6(i)
Date solved : Wednesday, March 05, 2025 at 09:17:36 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-2 x&=1 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 7.564 (sec). Leaf size: 14

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

\[ x \left (t \right ) = -\frac {1}{2}+\frac {3 \cosh \left (\sqrt {2}\, t \right )}{2} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 34

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

\[ x(t)\to \frac {1}{4} \left (3 e^{-\sqrt {2} t}+3 e^{\sqrt {2} t}-2\right ) \]

✓ Sympy. Time used: 0.131 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = \frac {3 e^{\sqrt {2} t}}{4} - \frac {1}{2} + \frac {3 e^{- \sqrt {2} t}}{4} \]