31.1.1 problem 10.2.4

Internal problem ID [7676]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page 307
Problem number : 10.2.4
Date solved : Tuesday, September 30, 2025 at 04:55:48 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-\omega ^{2} x&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

ode:=diff(diff(x(t),t),t)-omega^2*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = c_1 \,{\mathrm e}^{-\omega t}+c_2 \,{\mathrm e}^{\omega t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 23

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

\begin{align*} x(t)&\to c_1 e^{t \omega }+c_2 e^{-t \omega } \end{align*}

✓ Sympy. Time used: 0.043 (sec). Leaf size: 15

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

\[ x{\left (t \right )} = C_{1} e^{- \omega t} + C_{2} e^{\omega t} \]