76.29.6 problem 6

Internal problem ID [17882]
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 : 6
Date solved : Tuesday, January 28, 2025 at 11:09:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+6 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-2 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.103 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=5*x__1(t)+6*x__2(t)+2*x__3(t),diff(x__2(t),t)=-2*x__1(t)-2*x__2(t)-1*x__3(t),diff(x__3(t),t)=-2*x__1(t)-3*x__2(t)+0*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 75

DSolve[{D[x1[t],t]==5*x1[t]+6*x2[t]+2*x3[t],D[x2[t],t]==-2*x1[t]-2*x2[t]-1*x3[t],D[x3[t],t]==-2*x1[t]-3*x2[t]+0*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (4 c_1 t+6 c_2 t+2 c_3 t+c_1) \\ \text {x2}(t)\to e^t (c_2-(2 c_1+3 c_2+c_3) t) \\ \text {x3}(t)\to e^t (c_3-(2 c_1+3 c_2+c_3) t) \\ \end{align*}