76.23.1 problem 2

Internal problem ID [17792]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.1 (Definitions and examples). Problems at page 388
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 11:03:09 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_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 1.570 (sec). Leaf size: 75

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

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

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 164

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (2 e^{3 t}+1\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (6 t+4)-4\right )+c_2 \left (e^{3 t} (3 t+5)+4\right )+c_3 \left (e^{3 t} (3 t-4)+4\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (2-6 t)-2\right )+c_2 \left (2-e^{3 t} (3 t+2)\right )-c_3 \left (e^{3 t} (3 t-7)-2\right )\right ) \\ \end{align*}