76.19.6 problem 6

Internal problem ID [17730]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 11:01:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+29 y&={\mathrm e}^{-2 t} \sin \left (5 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 11.877 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+29*y(t)=exp(-2*t)*sin(5*t),y(0) = 5, D(y)(0) = -2],y(t), singsol=all)
 

\[ y = -\frac {{\mathrm e}^{-2 t} \left (-50+t \right ) \cos \left (5 t \right )}{10}+\frac {81 \,{\mathrm e}^{-2 t} \sin \left (5 t \right )}{50} \]

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 30

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+29*y[t]==Exp[-2*t]*Sin[5*t],{y[0]==5,Derivative[1][y][0] == -2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{50} e^{-2 t} (81 \sin (5 t)-5 (t-50) \cos (5 t)) \]