14.22.11 problem 11

Internal problem ID [2738]
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 : 11
Date solved : Monday, January 27, 2025 at 06:12:32 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 26

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

\begin{align*} \text {x1}(t)\to -2 e^{-2 t} \\ \text {x2}(t)\to 0 \\ \text {x3}(t)\to 3 e^{-2 t} \\ \end{align*}