28.4.33 problem 7.33

Internal problem ID [4565]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.33
Date solved : Monday, January 27, 2025 at 09:23:33 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 99

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

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