83.15.3 problem 4 (c)

Internal problem ID [22027]
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 (c)
Date solved : Thursday, October 02, 2025 at 08:21:48 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 )&=6 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 10

ode:=diff(diff(y(t),t),t)-4*y(t) = 0; 
ic:=[y(0) = 6, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 6 \cosh \left (2 t \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 11

ode=D[y[t],{t,2}]+4*y[t]==0; 
ic={y[0]==6,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 6 \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.035 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 6 \cos {\left (2 t \right )} \]