81.17.3 problem 21-10

Internal problem ID [21736]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 21. Applications of second order differential equations
Problem number : 21-10
Date solved : Thursday, October 02, 2025 at 08:01:34 PM
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 )&=3 \\ y^{\prime }\left (0\right )&=v \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+omega^2*y(x) = 0; 
ic:=[y(0) = 3, D(y)(0) = v]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {v \sin \left (\omega x \right )}{\omega }+3 \cos \left (\omega x \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

ode=D[y[x],{x,2}]+\[Omega]^2*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==v}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {v \sin (x \omega )}{\omega }+3 \cos (x \omega ) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
w = symbols("w") 
v = symbols("v") 
y = Function("y") 
ode = Eq(w**2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): v} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (- i v + 3 w\right ) e^{i w x}}{2 w} + \frac {\left (i v + 3 w\right ) e^{- i w x}}{2 w} \]