76.20.11 problem 11

Internal problem ID [17759]
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 : 11
Date solved : Tuesday, January 28, 2025 at 11:02:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=7 \end{align*}

✓ Solution by Maple

Time used: 12.434 (sec). Leaf size: 41

dsolve([diff(y(t),t$2)+4*y(t)=Heaviside(t-Pi)-Heaviside(t-3*Pi),y(0) = 3, D(y)(0) = 7],y(t), singsol=all)
 

\[ y = -\frac {\sin \left (t \right )^{2} \operatorname {Heaviside}\left (t -3 \pi \right )}{2}+\frac {\operatorname {Heaviside}\left (t -\pi \right ) \sin \left (t \right )^{2}}{2}+\frac {7 \sin \left (2 t \right )}{2}+3 \cos \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 48

DSolve[{D[y[t],{t,2}]+4*y[t]==UnitStep[t-Pi]-UnitStep[t-3*Pi],{y[0]==3,Derivative[1][y][0] ==7}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 \cos (2 t)+7 \cos (t) \sin (t) & t>3 \pi \lor t\leq \pi \\ \frac {1}{4} (11 \cos (2 t)+14 \sin (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]