52.6.9 problem 29

Internal problem ID [8344]
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 : 29
Date solved : Monday, January 27, 2025 at 03:49:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&={\mathrm e}^{t} \cos \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: 0.733 (sec). Leaf size: 24

dsolve([diff(y(t),t$2)-y(t)=exp(t)*cos(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-t}}{5}+\frac {{\mathrm e}^{t} \left (-\cos \left (t \right )+2 \sin \left (t \right )\right )}{5} \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 29

DSolve[{D[y[t],{t,2}]-y[t]==Exp[t]*Cos[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{5} \left (e^{-t}+2 e^t \sin (t)-e^t \cos (t)\right ) \]