64.20.7 problem 7

Internal problem ID [13650]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:54:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=18 \,{\mathrm e}^{-t} \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.955 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)-diff(y(t),t)-2*y(t)=18*exp(-t)*sin(3*t),y(0) = 0, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = {\mathrm e}^{-t} \left (\cos \left (3 t \right )-\sin \left (3 t \right )-3\right )+2 \,{\mathrm e}^{2 t} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 30

DSolve[{D[y[t],{t,2}]-D[y[t],t]-2*y[t]==18*Exp[-t]*Sin[3*t],{y[0]==0,Derivative[1][y][0]==3}},{y[t]},t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t} \left (2 e^{3 t}-\sin (3 t)+\cos (3 t)-3\right ) \]