14.18.6 problem 6

Internal problem ID [2690]
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 : 6
Date solved : Tuesday, March 04, 2025 at 02:34:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=\left \{\begin {array}{cc} \sin \left (2 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 )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 1.203 (sec). Leaf size: 88

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

\[ y = \frac {\left (\left \{\begin {array}{cc} {\mathrm e}^{-t} \left (35 t +29\right )-4 \cos \left (2 t \right )-3 \sin \left (2 t \right ) & t <\frac {\pi }{2} \\ \frac {35 \,{\mathrm e}^{-\frac {\pi }{2}} \pi }{2}+29 \,{\mathrm e}^{-\frac {\pi }{2}}+8 & t =\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \left (10 t -5 \pi +4\right )+{\mathrm e}^{-t} \left (35 t +29\right ) & \frac {\pi }{2}<t \end {array}\right .\right )}{25} \]

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 88

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} (t+1) & t\leq 0 \\ \frac {1}{25} e^{-t} \left (35 t+e^{\pi /2} (10 t-5 \pi +4)+29\right ) & 2 t>\pi \\ \frac {1}{25} e^{-t} \left (35 t-4 e^t \cos (2 t)-3 e^t \sin (2 t)+29\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

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