11.4.3 problem 3

Internal problem ID [1497]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:58:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.335 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)+4*y(t)=sin(t)-Heaviside(t-2*Pi)*sin(t-2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 27

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

\[ y(t)\to \frac {2}{3} \theta (2 \pi -t) \sin ^2\left (\frac {t}{2}\right ) \sin (t) \]