90.10.1 problem 17

Internal problem ID [25195]
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 149
Problem number : 17
Date solved : Thursday, October 02, 2025 at 11:57:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=4 \cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 10

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 10

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

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

✓ Sympy. Time used: 0.058 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 4*cos(2*t) + 5*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sqrt {5} \sin {\left (\frac {2 \sqrt {5} t}{5} \right )}}{2} - \frac {\cos {\left (2 t \right )}}{4} + \frac {\cos {\left (\frac {2 \sqrt {5} t}{5} \right )}}{4} \]