76.19.16 problem 16

Internal problem ID [17740]
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 : 16
Date solved : Tuesday, January 28, 2025 at 11:02:03 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-4 y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.097 (sec). Leaf size: 24

dsolve([diff(y__1(t),t) = 4*y__1(t)-4*y__2(t), diff(y__2(t),t) = 5*y__1(t)-4*y__2(t), y__1(0) = 1, y__2(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 25

DSolve[{D[y1[t],t]==4*y1[t]-4*y2[t],D[y2[t],t]==5*y1[t]-4*y2[t]},{y1[0]==1,y2[0]==0},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to 2 \sin (2 t)+\cos (2 t) \\ \text {y2}(t)\to 5 \sin (t) \cos (t) \\ \end{align*}