14.24.3 problem 1

Internal problem ID [2750]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 1
Date solved : Monday, January 27, 2025 at 06:12:43 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 99

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 (t+1)+(c_3-c_2) t) \\ \text {x2}(t)\to e^{-2 t} \left (c_1 \left (e^t (t+1)-1\right )-c_2 e^t t+c_3 e^t t+c_2\right ) \\ \text {x3}(t)\to e^{-2 t} \left (c_1 \left (e^t-1\right )-c_2 e^t+c_3 e^t+c_2\right ) \\ \end{align*}