14.22.9 problem 9

Internal problem ID [2736]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 9
Date solved : Monday, January 27, 2025 at 06:12:30 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 30

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

\begin{align*} \text {x1}(t)\to e^{2 t} \\ \text {x2}(t)\to -2 e^{2 t} \\ \text {x3}(t)\to -e^{2 t} \\ \end{align*}