76.15.21 problem 22

Internal problem ID [17662]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 22
Date solved : Tuesday, January 28, 2025 at 10:48:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=4*exp(-t)*cos(2*t),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.075 (sec). Leaf size: 31

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

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