76.29.11 problem 11

Internal problem ID [17887]
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 : 11
Date solved : Tuesday, January 28, 2025 at 11:09:45 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 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 )&=6 x_{1} \left (t \right )+4 x_{2} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-5 x_{1} \left (t \right )-2 x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2\\ x_{3} \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 45

dsolve([diff(x__1(t),t) = 4*x__1(t)+x__2(t)+3*x__3(t), diff(x__2(t),t) = 6*x__1(t)+4*x__2(t)+6*x__3(t), diff(x__3(t),t) = -5*x__1(t)-2*x__2(t)-4*x__3(t), x__1(0) = 1, x__2(0) = -2, x__3(0) = 5], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (16 t +1\right ) \\ x_{2} \left (t \right ) &= -32 \,{\mathrm e}^{t}+30 \,{\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= 15 \,{\mathrm e}^{t}-16 \,{\mathrm e}^{t} t -10 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 46

DSolve[{D[x1[t],t]==4*x1[t]+1*x2[t]+3*x3[t],D[x2[t],t]==6*x1[t]+4*x2[t]+6*x3[t],D[x3[t],t]==-5*x1[t]-2*x2[t]-4*x3[t]},{x1[0]==1,x2[0]==-2,x3[0]==5},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (16 t+1) \\ \text {x2}(t)\to 2 e^t \left (15 e^t-16\right ) \\ \text {x3}(t)\to e^t \left (-16 t-10 e^t+15\right ) \\ \end{align*}