20.16.6 problem 6

Internal problem ID [3839]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 6
Date solved : Monday, January 27, 2025 at 08:03:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 43

dsolve([diff(x__1(t),t)=2*x__1(t),diff(x__2(t),t)=5*x__2(t)-7*x__3(t),diff(x__3(t),t)=2*x__2(t)-4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+{\mathrm e}^{-2 t} c_{2} \\ x_{3} \left (t \right ) &= \frac {2 c_{1} {\mathrm e}^{3 t}}{7}+{\mathrm e}^{-2 t} c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 162

DSolve[{D[x1[t],t]==2*x1[t],D[x2[t],t]==5*x2[t]-7*x3[t],D[x3[t],t]==2*x2[t]-4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{2 t} \\ \text {x2}(t)\to \frac {1}{5} e^{-2 t} \left (c_2 \left (7 e^{5 t}-2\right )-7 c_3 \left (e^{5 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-2 t} \left (2 c_2 \left (e^{5 t}-1\right )+c_3 \left (7-2 e^{5 t}\right )\right ) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to \frac {1}{5} e^{-2 t} \left (c_2 \left (7 e^{5 t}-2\right )-7 c_3 \left (e^{5 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-2 t} \left (2 c_2 \left (e^{5 t}-1\right )+c_3 \left (7-2 e^{5 t}\right )\right ) \\ \end{align*}