63.7.4 problem 1(d)

Internal problem ID [13044]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.3 Complex eigenvalues. Exercises page 94
Problem number : 1(d)
Date solved : Wednesday, March 05, 2025 at 08:59:16 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-12 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)-12*x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 31

ode=D[x[t],{t,2}]-12*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.106 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-12*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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