52.6.17 problem 67

Internal problem ID [8352]
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 : Monday, January 27, 2025 at 03:49:23 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*}

✓ Solution by Maple

Time used: 0.712 (sec). Leaf size: 26

dsolve([diff(y(t),t$2)+4*y(t)=sin(t)*Heaviside(t-2*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 36

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

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