82.4.11 problem 28-11

Internal problem ID [21829]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-11
Date solved : Thursday, October 02, 2025 at 08:02:38 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t)+4*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \cos \left (2 t \right )+\sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 16

ode=D[y[t],{t,2}]+4*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \sin (2 t)+2 \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.035 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )} \]