67.4.28 problem Problem 4(d)

Internal problem ID [14001]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 4(d)
Date solved : Monday, March 31, 2025 at 08:21:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t & 0\le t <\pi \\ -t & \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: 0.368 (sec). Leaf size: 30

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

\[ y = \left \{\begin {array}{cc} -\sin \left (t \right )+t & t <\pi \\ -t -2 \pi \cos \left (t \right )-3 \sin \left (t \right ) & \pi \le t \end {array}\right . \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 38

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ t-\sin (t) & 0<t\leq \pi \\ -t-2 \pi \cos (t)-3 \sin (t) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.371 (sec). Leaf size: 12

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

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