20.20.7 problem 7

Internal problem ID [3897]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 7
Date solved : Monday, January 27, 2025 at 08:04:11 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 43

dsolve([diff(x__1(t),t)=3*x__1(t)-0*x__2(t)+4*x__3(t),diff(x__2(t),t)=0*x__1(t)+2*x__2(t)+0*x__3(t),diff(x__3(t),t)=-4*x__1(t)+0*x__2(t)-5*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 102

DSolve[{D[x1[t],t]==3*x1[t]-0*x2[t]+4*x3[t],D[x2[t],t]==0*x1[t]+2*x2[t]+0*x3[t],D[x3[t],t]==-4*x1[t]+0*x2[t]-5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} (4 c_1 t+4 c_2 t+c_1) \\ \text {x3}(t)\to e^{-t} (c_2-4 (c_1+c_2) t) \\ \text {x2}(t)\to c_3 e^{2 t} \\ \text {x1}(t)\to e^{-t} (4 c_1 t+4 c_2 t+c_1) \\ \text {x3}(t)\to e^{-t} (c_2-4 (c_1+c_2) t) \\ \text {x2}(t)\to 0 \\ \end{align*}