58.20.5 problem 5

Internal problem ID [14929]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:56:39 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y&=8 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+4*y(t) = 8; 
ic:=[y(0) = 0, D(y)(0) = 6]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2-2 \cos \left (2 t \right )+3 \sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 19

ode=D[y[t],{t,2}]+4*y[t]==8; 
ic={y[0]==0,Derivative[1][y][0]==6}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 \sin (2 t)-2 \cos (2 t)+2 \end{align*}

✓ Sympy. Time used: 0.044 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 8,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 3 \sin {\left (2 t \right )} - 2 \cos {\left (2 t \right )} + 2 \]