14.18.3 problem 3

Internal problem ID [2687]
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 : 3
Date solved : Sunday, March 30, 2025 at 12:14:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 1 & 0\le t <4 \\ 0 & 4<t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.330 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*y(t) = piecewise(0 <= t and t < 4,1,4 < t,0); 
ic:=y(0) = 3, D(y)(0) = -2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {11 \cos \left (2 t \right )}{4}-\sin \left (2 t \right )+\frac {\left (\left \{\begin {array}{cc} 1 & t <4 \\ \cos \left (2 t -8\right ) & 4\le t \end {array}\right .\right )}{4} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 72

ode=D[y[t],{t,2}]+4*y[t]==Piecewise[{{1,0<=t<4},{0,t>4}}]; 
ic={y[0]==3,Derivative[1][y][0] ==-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 \cos (2 t)-\sin (2 t) & t\leq 0 \\ \frac {1}{4} (11 \cos (2 t)-4 \sin (2 t)+1) & 0<t\leq 4 \\ \frac {1}{4} (\cos (8-2 t)+11 \cos (2 t)-4 \sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.345 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < 4)), (0, t > 4)) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {1}{4} & \text {for}\: t \geq 0 \wedge t < 4 \\0 & \text {for}\: t > 4 \\\text {NaN} & \text {otherwise} \end {cases} - \sin {\left (2 t \right )} + \frac {11 \cos {\left (2 t \right )}}{4} \]