64.20.17 problem 17

Internal problem ID [13589]
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 : 17
Date solved : Monday, March 31, 2025 at 08:02:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} -4 t +8 \pi & 0<t <2 \pi \\ 0 & 2<t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.317 (sec). Leaf size: 50

ode:=diff(diff(y(t),t),t)+4*y(t) = piecewise(0 < t and t < 2*Pi,-4*t+8*Pi,2 < t,0); 
ic:=y(0) = 2, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 56

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 \cos (2 t) & t\leq 0 \\ -2 (-1+\pi ) \cos (2 t) & t>2 \pi \\ \frac {1}{2} (-2 t-4 (-1+\pi ) \cos (2 t)+\sin (2 t)+4 \pi ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.590 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = C_{2} \cos {\left (2 t \right )} + \begin {cases} - t + 2 \pi & \text {for}\: t > 0 \wedge t < 2 \pi \\0 & \text {for}\: t > 2 \pi \\\text {NaN} & \text {otherwise} \end {cases} \]