76.20.11 problem 11

Internal problem ID [17688]
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 : 11
Date solved : Monday, March 31, 2025 at 04:24:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=7 \end{align*}

✓ Maple. Time used: 0.316 (sec). Leaf size: 41

ode:=diff(diff(y(t),t),t)+4*y(t) = Heaviside(t-Pi)-Heaviside(t-3*Pi); 
ic:=y(0) = 3, D(y)(0) = 7; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\sin \left (t \right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{2}-\frac {\sin \left (t \right )^{2} \operatorname {Heaviside}\left (t -3 \pi \right )}{2}+3 \cos \left (2 t \right )+\frac {7 \sin \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 48

ode=D[y[t],{t,2}]+4*y[t]==UnitStep[t-Pi]-UnitStep[t-3*Pi]; 
ic={y[0]==3,Derivative[1][y][0] ==7}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 \cos (2 t)+7 \cos (t) \sin (t) & t>3 \pi \lor t\leq \pi \\ \frac {1}{4} (11 \cos (2 t)+14 \sin (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.758 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Heaviside(t - 3*pi) - Heaviside(t - pi) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 7} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {\sin ^{2}{\left (t \right )} \theta \left (t - 3 \pi \right )}{2} + \frac {\sin ^{2}{\left (t \right )} \theta \left (t - \pi \right )}{2} + \frac {7 \sin {\left (2 t \right )}}{2} + 3 \cos {\left (2 t \right )} \]