76.20.1 problem 1

Internal problem ID [17678]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 1
Date solved : Monday, March 31, 2025 at 04:24:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 1 & 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 )&=5\\ y^{\prime }\left (0\right )&=3 \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 < 1/2*Pi,1,1/2*Pi <= t,0); 
ic:=y(0) = 5, D(y)(0) = 3; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 4 \cos \left (t \right )+\left (\left \{\begin {array}{cc} 1+3 \sin \left (t \right ) & t <\frac {\pi }{2} \\ 4 \sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 47

ode=D[y[t],{t,2}]+y[t]==Piecewise[{  {1,0<=t<Pi/2},{0,t>=Pi/2}}]; 
ic={y[0]==5,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 5 \cos (t)+3 \sin (t) & t\leq 0 \\ 4 \cos (t)+3 \sin (t)+1 & t>0\land 2 t\leq \pi \\ 4 (\cos (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.291 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t <= pi/2)), (0, t >= pi/2)) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} 1 & \text {for}\: t \geq 0 \wedge t \leq \frac {\pi }{2} \\0 & \text {for}\: t > \frac {\pi }{2} \\\text {NaN} & \text {otherwise} \end {cases} + 3 \sin {\left (t \right )} + 4 \cos {\left (t \right )} \]