76.18.11 problem 23

Internal problem ID [17651]
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 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 23
Date solved : Monday, March 31, 2025 at 04:23:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=9\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.289 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)+16*y(t) = piecewise(0 <= t and t < Pi,1,Pi <= t,0); 
ic:=y(0) = 9, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\sin \left (4 t \right )}{2}+\left (\left \{\begin {array}{cc} \frac {1}{16}+\frac {143 \cos \left (4 t \right )}{16} & t <\pi \\ 9 \cos \left (4 t \right ) & \pi \le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 41

ode=D[y[t],{t,2}]+16*y[t]==Piecewise[{  {1,0<= t <Pi},{0,t >= Pi}}]; 
ic={y[0]==0,Derivative[1][y][0] == 2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} \sin (4 t) & t>\pi \lor t\leq 0 \\ \frac {1}{16} (-\cos (4 t)+8 \sin (4 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.394 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < pi)), (0, t > pi)) + 16*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {1}{16} & \text {for}\: t \geq 0 \wedge t < \pi \\0 & \text {for}\: t > \pi \\\text {NaN} & \text {otherwise} \end {cases} + \frac {\sin {\left (4 t \right )}}{2} - \frac {\cos {\left (4 t \right )}}{16} \]