90.8.14 problem 14

Internal problem ID [25188]
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 : 14
Date solved : Thursday, October 02, 2025 at 11:57:40 PM
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 )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 12

ode:=diff(diff(y(t),t),t)+4*y(t) = 8; 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 12

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

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

✓ Sympy. Time used: 0.043 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 8,0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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