76.19.14 problem 14

Internal problem ID [17738]
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 : 14
Date solved : Tuesday, January 28, 2025 at 11:02:01 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=-5 y_{1} \left (t \right )+y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-9 y_{1} \left (t \right )+5 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: 2.509 (sec). Leaf size: 33

dsolve([diff(y__1(t),t) = -5*y__1(t)+y__2(t), diff(y__2(t),t) = -9*y__1(t)+5*y__2(t), y__1(0) = 1, y__2(0) = 0], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= \frac {9 \,{\mathrm e}^{-4 t}}{8}-\frac {{\mathrm e}^{4 t}}{8} \\ y_{2} \left (t \right ) &= \frac {9 \,{\mathrm e}^{-4 t}}{8}-\frac {9 \,{\mathrm e}^{4 t}}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 40

DSolve[{D[y1[t],t]==-5*y1[t]+y2[t],D[y2[t],t]==-9*y1[t]+5*y2[t]},{y1[0]==1,y2[0]==0},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to -\frac {1}{8} e^{-4 t} \left (e^{8 t}-9\right ) \\ \text {y2}(t)\to -\frac {9}{8} e^{-4 t} \left (e^{8 t}-1\right ) \\ \end{align*}