20.16.14 problem 14

Internal problem ID [3847]
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 : 14
Date solved : Monday, January 27, 2025 at 08:03:21 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 40

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 119

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

\begin{align*} \text {x1}(t)\to \frac {1}{4} \left (2 c_1 \left (e^{4 t}+1\right )-(c_2-3 c_3) \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{4} \left (2 c_1 \left (e^{4 t}-1\right )-c_2 \left (e^{4 t}-5\right )+3 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{4} \left (2 c_1 \left (e^{4 t}-1\right )-c_2 e^{4 t}+3 c_3 e^{4 t}+c_2+c_3\right ) \\ \end{align*}