28.4.38 problem 7.38

Internal problem ID [4570]
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 : 7.38
Date solved : Monday, January 27, 2025 at 09:23:38 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 73

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 120

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

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