13.10.5 problem 5

Internal problem ID [2406]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.4, The method of variation of parameters. Page 154
Problem number : 5
Date solved : Monday, January 27, 2025 at 05:51:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 3 y^{\prime \prime }+4 y^{\prime }+y&=\sin \left (t \right ) {\mathrm e}^{-t} \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: 0.024 (sec). Leaf size: 27

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

\[ y = \frac {24 \,{\mathrm e}^{-\frac {t}{3}}}{13}+\frac {\left (-13+2 \cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{13} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 33

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

\[ y(t)\to \frac {1}{13} e^{-t} \left (24 e^{2 t/3}-3 \sin (t)+2 \cos (t)-13\right ) \]