14.24.5 problem 3

Internal problem ID [2752]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:12:45 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_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 33

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

\begin{align*} x_{1} \left (t \right ) &= \left (-c_2 t +c_1 \right ) {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 74

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1-c_2 t) \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x3}(t)\to c_3 e^{-2 t} \\ \text {x1}(t)\to e^{-t} (c_1-c_2 t) \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x3}(t)\to 0 \\ \end{align*}