76.19.23 problem 23

Internal problem ID [17747]
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 : 23
Date solved : Tuesday, January 28, 2025 at 11:02:08 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.097 (sec). Leaf size: 49

dsolve([diff(y__1(t),t) = -2*y__1(t)+y__2(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)+sin(t), y__1(0) = 0, y__2(0) = 0], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-3 t}}{20}+\frac {{\mathrm e}^{-t}}{4}-\frac {\cos \left (t \right )}{5}+\frac {\sin \left (t \right )}{10} \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t}}{20}+\frac {{\mathrm e}^{-t}}{4}+\frac {2 \sin \left (t \right )}{5}-\frac {3 \cos \left (t \right )}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.096 (sec). Leaf size: 60

DSolve[{D[y1[t],t]==-2*y1[t]+1*y2[t]+0,D[y2[t],t]==1*y1[t]-2*y2[t]+Sin[t]},{y1[0]==0,y2[0]==0},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{20} \left (-e^{-3 t}+5 e^{-t}+2 \sin (t)-4 \cos (t)\right ) \\ \text {y2}(t)\to \frac {1}{20} \left (e^{-3 t}+5 e^{-t}+8 \sin (t)-6 \cos (t)\right ) \\ \end{align*}