90.10.4 problem 20

Internal problem ID [25198]
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 : 20
Date solved : Thursday, October 02, 2025 at 11:57:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.107 (sec). Leaf size: 15

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

\[ y = \sin \left (t \right ) \left (1-{\mathrm e}^{-t}\right ) \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 17

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

\begin{align*} y(t)&\to \left (1-e^{-t}\right ) \sin (t) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 12

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

\[ y{\left (t \right )} = \sin {\left (t \right )} - e^{- t} \sin {\left (t \right )} \]