76.21.9 problem 9

Internal problem ID [17773]
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 5.7 (Impulse Functions). Problems at page 350
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 11:02:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+3 \delta \left (t -\frac {3 \pi }{2}\right )-\operatorname {Heaviside}\left (t -2 \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*}

✓ Solution by Maple

Time used: 13.088 (sec). Leaf size: 39

dsolve([diff(y(t),t$2)+y(t)=Heaviside(t-Pi/2)+3*Dirac(t-3*Pi/2)-Heaviside(t-2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.331 (sec). Leaf size: 75

DSolve[{D[y[t],{t,2}]+y[t]==UnitStep[t-Pi/2]+3*DiracDelta[t-3*Pi/2]-UnitStep[t-2*Pi],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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