76.18.8 problem 19

Internal problem ID [17719]
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 : 19
Date solved : Tuesday, January 28, 2025 at 11:01:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=t \cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.457 (sec). Leaf size: 37

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=t*cos(2*t),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\cos \left (2 t \right ) \left (-2+17 t +291 \,{\mathrm e}^{-t}\right )}{289}+\frac {4 \sin \left (2 t \right ) \left (t +\frac {213 \,{\mathrm e}^{-t}}{68}-\frac {19}{17}\right )}{17} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 47

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

\[ y(t)\to \frac {1}{289} e^{-t} \left (\left (e^t (68 t-76)+213\right ) \sin (2 t)+\left (e^t (17 t-2)+291\right ) \cos (2 t)\right ) \]