28.4.48 problem 7.48

Internal problem ID [4580]
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 : 7.48
Date solved : Monday, January 27, 2025 at 09:23:46 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 )&=x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )+6 \,{\mathrm e}^{-t}\\ x_{3}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 69

dsolve([diff(x__1(t),t)=x__1(t)-x__2(t)+x__3(t),diff(x__2(t),t)=x__1(t)+x__2(t)-x__3(t)+6*exp(-t),diff(x__3(t),t)=2*x__1(t)-x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} c_{1} -\frac {c_{2} {\mathrm e}^{-t}}{5}+{\mathrm e}^{2 t} c_3 -\frac {6 \,{\mathrm e}^{-t}}{5} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} c_{1} +\frac {3 c_{2} {\mathrm e}^{-t}}{5}-\frac {12 \,{\mathrm e}^{-t}}{5} \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{-t}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 177

DSolve[{D[x1[t],t]==x1[t]-x2[t]+x3[t],D[x2[t],t]==x1[t]+x2[t]-x3[t]+6*Exp[-t],D[x3[t],t]==2*x1[t]-x2[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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