52.6.19 problem 69

Internal problem ID [8354]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 69
Date solved : Wednesday, March 05, 2025 at 05:35:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 1 & \pi \le t <2 \pi \\ 0 & 2 \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 )&=1 \end{align*}

✓ Maple. Time used: 1.014 (sec). Leaf size: 30

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

\[ y = \sin \left (t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \cos \left (t \right )+1 & t <2 \pi \\ 2 \cos \left (t \right ) & 2 \pi \le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 35

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

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

✓ Sympy. Time used: 0.472 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((0, (t >= 0) & (t < pi)), (1, (t >= pi) & (t < 2*pi)), (0, t >= 2*pi)) + y(t) + 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)
 

\[ y{\left (t \right )} = \begin {cases} 0 & \text {for}\: t \geq 0 \wedge t < \pi \\1 & \text {for}\: t \geq \pi \wedge t < 2 \pi \\0 & \text {for}\: t \geq 2 \pi \\\text {NaN} & \text {otherwise} \end {cases} + \sin {\left (t \right )} \]