76.19.17 problem 17

Internal problem ID [17741]
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 : 17
Date solved : Tuesday, January 28, 2025 at 11:02:04 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=6 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-6 y_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 5\\ y_{2} \left (0\right ) = 4 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 33

dsolve([diff(y__1(t),t) = 6*y__2(t), diff(y__2(t),t) = -6*y__1(t), y__1(0) = 5, y__2(0) = 4], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= 4 \sin \left (6 t \right )+5 \cos \left (6 t \right ) \\ y_{2} \left (t \right ) &= 4 \cos \left (6 t \right )-5 \sin \left (6 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 34

DSolve[{D[y1[t],t]==0*y1[t]+6*y2[t],D[y2[t],t]==-6*y1[t]+0*y2[t]},{y1[0]==5,y2[0]==4},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to 4 \sin (6 t)+5 \cos (6 t) \\ \text {y2}(t)\to 4 \cos (6 t)-5 \sin (6 t) \\ \end{align*}