46.6.17 problem 67

Internal problem ID [9638]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 67
Date solved : Tuesday, September 30, 2025 at 06:21:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.353 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*y(t) = sin(t)*Heaviside(t-2*Pi); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\left (\cos \left (t \right )-1\right ) \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )}{3}+2 \cos \left (t \right )^{2}-1 \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 36

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t) & t\leq 2 \pi \\ \frac {1}{3} (3 \cos (2 t)-\cos (t) \sin (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 1.079 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - sin(t)*Heaviside(t - 2*pi) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\sin ^{3}{\left (t \right )} \theta \left (t - 2 \pi \right )}{3} + 1\right ) \cos {\left (2 t \right )} + \left (\frac {\cos {\left (t \right )} \theta \left (t - 2 \pi \right )}{4} - \frac {\cos {\left (3 t \right )} \theta \left (t - 2 \pi \right )}{12} - \frac {\theta \left (t - 2 \pi \right )}{6}\right ) \sin {\left (2 t \right )} \]