63.15.3 problem 6(c)

Internal problem ID [13115]
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(c)
Date solved : Monday, March 31, 2025 at 07:34:38 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-x^{\prime }-6 x&=0 \end{align*}

Using Laplace method With initial conditions

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

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

ode:=diff(diff(x(t),t),t)-diff(x(t),t)-6*x(t) = 0; 
ic:=x(0) = 2, D(x)(0) = -1; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = \frac {7 \,{\mathrm e}^{-2 t}}{5}+\frac {3 \,{\mathrm e}^{3 t}}{5} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 23

ode=D[x[t],{t,2}]-D[x[t],t]-6*x[t]==0; 
ic={x[0]==2,Derivative[1][x][0 ]==-1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{5} e^{-2 t} \left (3 e^{5 t}+7\right ) \]

✓ Sympy. Time used: 0.190 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-6*x(t) - Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 2, Subs(Derivative(x(t), t), t, 0): -1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {3 e^{3 t}}{5} + \frac {7 e^{- 2 t}}{5} \]