76.19.22 problem 22

Internal problem ID [17746]
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 : 22
Date solved : Tuesday, January 28, 2025 at 11:02:07 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-5 y_{2} \left (t \right )+3\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )+3 y_{2} \left (t \right )+5 \cos \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.325 (sec). Leaf size: 51

dsolve([diff(y__1(t),t) = -y__1(t)-5*y__2(t)+3, diff(y__2(t),t) = y__1(t)+3*y__2(t)+5*cos(t), y__1(0) = 1, y__2(0) = -1], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= -\frac {27 \,{\mathrm e}^{t} \sin \left (t \right )}{2}+\frac {21 \,{\mathrm e}^{t} \cos \left (t \right )}{2}+10 \sin \left (t \right )-5 \cos \left (t \right )-\frac {9}{2} \\ y_{2} \left (t \right ) &= \frac {15 \,{\mathrm e}^{t} \sin \left (t \right )}{2}-\frac {3 \,{\mathrm e}^{t} \cos \left (t \right )}{2}-\cos \left (t \right )-3 \sin \left (t \right )+\frac {3}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.113 (sec). Leaf size: 62

DSolve[{D[y1[t],t]==-1*y1[t]-5*y2[t]+3,D[y2[t],t]==1*y1[t]+3*y2[t]+5*Cos[t]},{y1[0]==1,y2[0]==-1},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{2} \left (\left (20-27 e^t\right ) \sin (t)+\left (21 e^t-10\right ) \cos (t)-9\right ) \\ \text {y2}(t)\to \frac {1}{2} \left (3 \left (5 e^t-2\right ) \sin (t)-\left (\left (3 e^t+2\right ) \cos (t)\right )+3\right ) \\ \end{align*}