9.18.39 problem 39

Internal problem ID [3282]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 06:32:21 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 13

ode:=diff(diff(x(t),t),t)-k^2*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = v__0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {v_{0} \sinh \left (k t \right )}{k} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 27

ode=D[x[t],{t,2}]-k^2*x[t]==0; 
ic={x[0]==0,Derivative[1][x][0 ]==v0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {\text {v0} e^{-k t} \left (e^{2 k t}-1\right )}{2 k} \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
k = symbols("k") 
v__0 = symbols("v__0") 
x = Function("x") 
ode = Eq(-k**2*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): v__0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {v^{0} e^{k t}}{2 k} - \frac {v^{0} e^{- k t}}{2 k} \]