76.29.4 problem 4

Internal problem ID [17880]
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 : 4
Date solved : Tuesday, January 28, 2025 at 11:09:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-5 x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-4 x_{1} \left (t \right )+3 x_{2} \left (t \right )+3 x_{3} \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)-3*x__2(t)-2*x__3(t),diff(x__2(t),t)=8*x__1(t)-5*x__2(t)-4*x__3(t),diff(x__3(t),t)=-4*x__1(t)+3*x__2(t)+3*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 77

DSolve[{D[x1[t],t]==5*x1[t]-3*x2[t]-2*x3[t],D[x2[t],t]==8*x1[t]-5*x2[t]-4*x3[t],D[x3[t],t]==-4*x1[t]+3*x2[t]+3*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (4 c_1 t-3 c_2 t-2 c_3 t+c_1) \\ \text {x2}(t)\to e^t ((8 c_1-6 c_2-4 c_3) t+c_2) \\ \text {x3}(t)\to e^t ((-4 c_1+3 c_2+2 c_3) t+c_3) \\ \end{align*}