76.29.3 problem 3

Internal problem ID [17879]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:09:37 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 72

dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)+1*x__3(t),diff(x__2(t),t)=2*x__1(t)+1*x__2(t)-x__3(t),diff(x__3(t),t)=-3*x__1(t)+2*x__2(t)+4*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 110

DSolve[{D[x1[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[x2[t],t]==2*x1[t]+1*x2[t]-x3[t],D[x3[t],t]==-3*x1[t]+2*x2[t]+4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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