76.20.4 problem 4

Internal problem ID [17752]
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 : Tuesday, January 28, 2025 at 11:02:13 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sin \left (t \right )-\operatorname {Heaviside}\left (t -\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: 11.267 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 24

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

\[ y(t)\to -\frac {1}{3} \sin (t) (\theta (\pi -t) (\cos (t)+1)-2) \]