90.8.10 problem 10

Internal problem ID [25184]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 109
Problem number : 10
Date solved : Thursday, October 02, 2025 at 11:57:38 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+a^2*y(t) = 0; 
ic:=[y(0) = y__0, D(y)(0) = y__1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = y_{0} \cos \left (a t \right )+\frac {y_{1} \sin \left (a t \right )}{a} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+a^2*y[t]==0; 
ic={y[0]==y0,Derivative[1][y][0] ==y1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \text {y0} \cos (a t)+\frac {\text {y1} \sin (a t)}{a} \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y0 = symbols("y0") 
y1 = symbols("y1") 
y = Function("y") 
ode = Eq(a**2*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): y0, Subs(Derivative(y(t), t), t, 0): y1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\left (a y_{0} - i y_{1}\right ) e^{i a t}}{2 a} + \frac {\left (a y_{0} + i y_{1}\right ) e^{- i a t}}{2 a} \]