52.7.4 problem 12

Internal problem ID [8359]
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. 7.4.1 DERIVATIVES OF A TRANSFORM. Page 309
Problem number : 12
Date solved : Monday, January 27, 2025 at 03:49:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.577 (sec). Leaf size: 16

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

\[ y = -\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right ) \left (t -2\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 21

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

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