14.18.4 problem 4

Internal problem ID [2688]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.11, Differential equations with discontinuous right-hand sides. Excercises page 243
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:14:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ \cos \left (t \right ) & \pi \le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.349 (sec). Leaf size: 40

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

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

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 54

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

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

✓ Sympy. Time used: 0.584 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((sin(t), (t >= 0) & (t < pi)), (cos(t), t >= pi)) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {\sin {\left (t \right )}}{3} & \text {for}\: t \geq 0 \wedge t < \pi \\\frac {\cos {\left (t \right )}}{3} & \text {for}\: t \geq \pi \\\text {NaN} & \text {otherwise} \end {cases} - \frac {\sin {\left (2 t \right )}}{6} + \cos {\left (2 t \right )} \]