28.4.37 problem 7.37

Internal problem ID [4569]
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 : 7.37
Date solved : Monday, January 27, 2025 at 09:23:37 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 )\\ x_{3}^{\prime }\left (t \right )&=-2 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2\\ x_{3} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 43

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), diff(x__3(t),t) = -2*x__2(t)+2*x__3(t), x__1(0) = 1, x__2(0) = -2, x__3(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 48

DSolve[{D[x1[t],t]==x1[t]-x2[t]+x3[t],D[x2[t],t]==x1[t]+x2[t]-x3[t],D[x3[t],t]==-2*x2[t]+2*x3[t]},{x1[0]==1,x2[0]==-2,x3[0]==0},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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