14.24.7 problem 5

Internal problem ID [2754]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:12:46 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 15

dsolve([diff(x__1(t),t) = -x__1(t)+x__2(t)+2*x__3(t), diff(x__2(t),t) = -x__1(t)+x__2(t)+x__3(t), diff(x__3(t),t) = -2*x__1(t)+x__2(t)+3*x__3(t), x__1(0) = 1, x__2(0) = 0, x__3(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= 0 \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 18

DSolve[{D[ x1[t],t]==-1*x1[t]+1*x2[t]+2*x3[t],D[ x2[t],t]==-1*x1[t]+1*x2[t]+1*x3[t],D[ x3[t],t]==-2*x1[t]+1*x2[t]+3*x3[t]},{x1[0]==1,x2[0]==0,x3[0]==1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t \\ \text {x2}(t)\to 0 \\ \text {x3}(t)\to e^t \\ \end{align*}