70.20.4 problem 4

Internal problem ID [19038]
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 : 4
Date solved : Thursday, October 02, 2025 at 03:37:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.330 (sec). Leaf size: 23

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

\[ y = -\frac {\sin \left (t \right ) \left (\left (\cos \left (t \right )+1\right ) \operatorname {Heaviside}\left (t -\pi \right )+\cos \left (t \right )-1\right )}{3} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 24

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

\begin{align*} y(t)&\to -\frac {1}{3} \sin (t) (\theta (\pi -t) (\cos (t)+1)-2) \end{align*}

✓ Sympy. Time used: 1.972 (sec). Leaf size: 92

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

\[ y{\left (t \right )} = - \frac {2 \sin ^{5}{\left (t \right )} \theta \left (t - \pi \right )}{3} + \frac {5 \sin ^{3}{\left (t \right )} \theta \left (t - \pi \right )}{6} + \frac {\sin {\left (t \right )} \cos {\left (t \right )} \cos {\left (3 t \right )} \theta \left (t - \pi \right )}{6} - \frac {\sin {\left (t \right )} \cos {\left (t \right )} \theta \left (t - \pi \right )}{3} - \frac {\sin {\left (t \right )} \theta \left (t - \pi \right )}{2} + \frac {\sin {\left (t \right )}}{3} - \frac {\sin {\left (2 t \right )}}{6} \]