65.6.15 problem 12.1 (xv)

Internal problem ID [13693]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 12, Homogeneous second order linear equations. Exercises page 118
Problem number : 12.1 (xv)
Date solved : Monday, March 31, 2025 at 08:09:09 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\omega ^{2} y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 12

ode:=diff(diff(y(t),t),t)+omega^2*y(t) = 0; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {\sin \left (\omega t \right )}{\omega } \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 13

ode=D[y[t],{t,2}]+w^2*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {\sin (t w)}{w} \]

✓ Sympy. Time used: 0.109 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = - \frac {i e^{i \omega t}}{2 \omega } + \frac {i e^{- i \omega t}}{2 \omega } \]