20.16.17 problem 17

Internal problem ID [3850]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 17
Date solved : Monday, January 27, 2025 at 08:03:24 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 3\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 25

dsolve([diff(x__1(t),t) = -x__1(t)+4*x__2(t), diff(x__2(t),t) = 2*x__1(t)-3*x__2(t), x__1(0) = 3, x__2(0) = 0], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{t}+{\mathrm e}^{-5 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t}-{\mathrm e}^{-5 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 30

DSolve[{D[x1[t],t]==-x1[t]+4*x2[t],D[x2[t],t]==2*x1[t]-3*x2[t]},{x1[0]==3,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-5 t}+2 e^t \\ \text {x2}(t)\to e^t-e^{-5 t} \\ \end{align*}