83.15.5 problem 4 (e)

Internal problem ID [22029]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 4 (e)
Date solved : Thursday, October 02, 2025 at 08:21:48 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)+n^2*y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = k]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

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

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

\begin{align*} y(t)&\to \frac {k \sin (n t)}{n} \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 29

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

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