82.4.18 problem 28-18

Internal problem ID [21836]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-18
Date solved : Thursday, October 02, 2025 at 08:02:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t \end {array}\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: 0.188 (sec). Leaf size: 66

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = piecewise(0 <= t and t < Pi,1,Pi <= t,0); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 1+\frac {\left (-\sin \left (2 t \right )-2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}}{2} & t <\pi \\ 2-{\mathrm e}^{-\pi } & t =\pi \\ \frac {\left (\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t} \left (-1+{\mathrm e}^{\pi }\right )}{2} & \pi <t \end {array}\right .\right )}{5} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 71

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==Piecewise[{{1,0<=t<Pi},{0,t>=Pi}}]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {1}{10} e^{-t} \left (-2 \cos (2 t)+2 e^t-\sin (2 t)\right ) & 0<t\leq \pi \\ \frac {1}{10} e^{-t} \left (-1+e^{\pi }\right ) (2 \cos (2 t)+\sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < pi)), (0, t >= pi)) + 5*y(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)