20.13.9 problem 9

Internal problem ID [3818]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 9
Date solved : Monday, January 27, 2025 at 08:02:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+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: 0.023 (sec). Leaf size: 29

dsolve([diff(x__1(t),t) = 2*x__2(t), diff(x__2(t),t) = x__1(t)+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}^{2 t} \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-t}+{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 34

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

\begin{align*} \text {x1}(t)\to e^{-t} \left (e^{3 t}+2\right ) \\ \text {x2}(t)\to e^{-t} \left (e^{3 t}-1\right ) \\ \end{align*}