28.4.49 problem 7.49

Internal problem ID [4581]
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 : 7.49
Date solved : Monday, January 27, 2025 at 09:23:47 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 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 )+12 t\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 68

dsolve([diff(x__1(t),t)=x__1(t)-2*x__2(t)-x__3(t),diff(x__2(t),t)=-x__1(t)+x__2(t)+x__3(t)+12*t,diff(x__3(t),t)=x__1(t)-x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.465 (sec). Leaf size: 160

DSolve[{D[x1[t],t]==x1[t]-2*x2[t]-x3[t],D[x2[t],t]==-x1[t]+x2[t]+x3[t]+12*t,D[x3[t],t]==x1[t]-x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} \left (12 t^2+12 t+c_1 \left (e^{2 t}+1\right )-2 c_2 e^{2 t}-c_3 e^{2 t}+6+2 c_2+c_3\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (-18 e^t+(-c_1+2 c_2+c_3) e^{3 t}+c_1+c_2-c_3\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-t} \left (3 e^t \left (12 t^2-12 t+18+c_1+2 c_2+c_3\right )+(c_1-2 c_2-c_3) e^{3 t}-4 (c_1+c_2-c_3)\right ) \\ \end{align*}