28.4.54 problem 7.54

Internal problem ID [4586]
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 : 7.54
Date solved : Monday, January 27, 2025 at 09:25:45 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 )+2 \,{\mathrm e}^{2 t}\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )-3 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: 0.095 (sec). Leaf size: 60

dsolve([diff(x__1(t),t)=2*x__1(t)-x__2(t)-x__3(t)+2*exp(2*t),diff(x__2(t),t)=3*x__1(t)-2*x__2(t)-3*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 ) &= \frac {2 c_{2} {\mathrm e}^{t}}{3}+3 \,{\mathrm e}^{2 t}+\frac {c_3}{3}+{\mathrm e}^{t} c_{1} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{t}+3 \,{\mathrm e}^{2 t}+c_3 \\ x_{3} \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{t}}{3}-{\mathrm e}^{2 t}-\frac {c_3}{3}+{\mathrm e}^{t} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 119

DSolve[{D[x1[t],t]==2*x1[t]-x2[t]-x3[t]+2*Exp[2*t],D[x2[t],t]==3*x1[t]-2*x2[t]-3*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 3 e^{2 t}+(-6+2 c_1-c_2-c_3) e^t+2-c_1+c_2+c_3 \\ \text {x2}(t)\to 3 e^{2 t}+(-9+3 c_1-2 c_2-3 c_3) e^t+3 (2-c_1+c_2+c_3) \\ \text {x3}(t)\to -e^{2 t}+(3-c_1+c_2+2 c_3) e^t-2+c_1-c_2-c_3 \\ \end{align*}