14.18.5 problem 5

Internal problem ID [2689]
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 : 5
Date solved : Sunday, March 30, 2025 at 12:14:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\frac {\pi }{2} \\ 0 & \frac {\pi }{2}\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.332 (sec). Leaf size: 40

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(0 <= t and t < 1/2*Pi,cos(t),1/2*Pi <= t,0); 
ic:=y(0) = 3, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 6 \cos \left (t \right )+\sin \left (t \right ) \left (-2+t \right ) & t <\frac {\pi }{2} \\ \frac {\left (\pi -4\right ) \sin \left (t \right )}{2}+5 \cos \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right .\right )}{2} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 55

ode=D[y[t],{t,2}]+y[t]==Piecewise[{{Cos[t],0<=t<Pi/2},{0,t>=Pi/2}}]; 
ic={y[0]==3,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 \cos (t)-\sin (t) & t\leq 0 \\ \frac {1}{4} (10 \cos (t)+(-4+\pi ) \sin (t)) & 2 t>\pi \\ 3 \cos (t)+\frac {1}{2} (t-2) \sin (t) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.392 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((cos(t), (t >= 0) & (t < pi/2)), (0, t >= pi/2)) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {t \sin {\left (t \right )}}{2} + \frac {\cos {\left (t \right )}}{4} & \text {for}\: t \geq 0 \wedge t < \frac {\pi }{2} \\0 & \text {for}\: t \geq \frac {\pi }{2} \\\text {NaN} & \text {otherwise} \end {cases} - \sin {\left (t \right )} + \frac {11 \cos {\left (t \right )}}{4} \]