76.18.7 problem 18

Internal problem ID [17718]
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 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 11:01:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 \,{\mathrm e}^{-2 t} \sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 12.494 (sec). Leaf size: 35

dsolve([diff(y(t),t$2)+4*y(t)=3*exp(-2*t)*sin(2*t),y(0) = 2, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = \frac {\left (6 \,{\mathrm e}^{-2 t}+34\right ) \cos \left (2 t \right )}{20}+\frac {3 \sin \left (2 t \right ) {\mathrm e}^{-2 t}}{20}-\frac {7 \sin \left (2 t \right )}{20} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 44

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

\[ y(t)\to \frac {1}{20} e^{-2 t} \left (\left (34 e^{2 t}+6\right ) \cos (2 t)-\left (7 e^{2 t}-3\right ) \sin (2 t)\right ) \]