28.4.36 problem 7.36

Internal problem ID [4568]
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 : 7.36
Date solved : Monday, January 27, 2025 at 09:23:36 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )-x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 66

dsolve([diff(x__1(t),t)=3*x__1(t)-2*x__2(t)-x__3(t),diff(x__2(t),t)=3*x__1(t)-4*x__2(t)-3*x__3(t),diff(x__3(t),t)=2*x__1(t)-4*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-5 t}}{2}+\frac {{\mathrm e}^{2 t} c_3}{2}+{\mathrm e}^{2 t} c_{1} \\ x_{2} \left (t \right ) &= \frac {3 c_{2} {\mathrm e}^{-5 t}}{2}-\frac {{\mathrm e}^{2 t} c_3}{4}+\frac {{\mathrm e}^{2 t} c_{1}}{2} \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{-5 t}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 135

DSolve[{D[x1[t],t]==3*x1[t]-2*x2[t]-x3[t],D[x2[t],t]==3*x1[t]-4*x2[t]-3*x3[t],D[x3[t],t]==2*x1[t]-4*x2[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{7} e^{-5 t} \left (c_1 \left (8 e^{7 t}-1\right )-(2 c_2+c_3) \left (e^{7 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{7} e^{-5 t} \left (3 c_1 \left (e^{7 t}-1\right )+c_2 \left (e^{7 t}+6\right )-3 c_3 \left (e^{7 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{7} e^{-5 t} \left (2 c_1 \left (e^{7 t}-1\right )-4 c_2 \left (e^{7 t}-1\right )+c_3 \left (5 e^{7 t}+2\right )\right ) \\ \end{align*}