14.22.10 problem 10

Internal problem ID [2737]
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 : 10
Date solved : Monday, January 27, 2025 at 06:12:31 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+10 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -1\\ x_{2} \left (0\right ) = -4\\ x_{3} \left (0\right ) = 13 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 36

dsolve([diff(x__1(t),t) = x__1(t)-x__2(t), diff(x__2(t),t) = x__1(t)+2*x__2(t)+x__3(t), diff(x__3(t),t) = x__1(t)+10*x__2(t)+2*x__3(t), x__1(0) = -1, x__2(0) = -4, x__3(0) = 13], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t}-2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -4 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= -{\mathrm e}^{t}+14 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 42

DSolve[{D[ x1[t],t]==1*x1[t]-1*x2[t]-0*x3[t],D[ x2[t],t]==1*x1[t]+2*x2[t]+1*x3[t],D[ x3[t],t]==1*x1[t]+10*x2[t]+2*x3[t]},{x1[0]==-1,x2[0]==-4,x3[0]==13},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t-2 e^{-t} \\ \text {x2}(t)\to -4 e^{-t} \\ \text {x3}(t)\to 14 e^{-t}-e^t \\ \end{align*}