89.13.19 problem 19

Internal problem ID [24602]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:46:25 PM
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: 0.011 (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);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 14

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} \sin (k t)}{k} \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
v0 = symbols("v0") 
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): v0} 
dsolve(ode,func=x(t),ics=ics)
 

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