76.25.5 problem 5

Internal problem ID [17818]
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 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:03:23 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.124 (sec). Leaf size: 63

dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)+6*x__3(t),diff(x__2(t),t)=1*x__1(t)+6*x__2(t)+1*x__3(t),diff(x__3(t),t)=6*x__1(t)+1*x__2(t)+1*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{5 t}+c_{2} {\mathrm e}^{-5 t}+c_{3} {\mathrm e}^{8 t} \\ x_{2} \left (t \right ) &= -2 c_{1} {\mathrm e}^{5 t}+c_{3} {\mathrm e}^{8 t} \\ x_{3} \left (t \right ) &= c_{1} {\mathrm e}^{5 t}-c_{2} {\mathrm e}^{-5 t}+c_{3} {\mathrm e}^{8 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 163

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

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{-5 t} \left (c_1 \left (e^{10 t}+2 e^{13 t}+3\right )+(c_3-2 c_2) e^{10 t}+2 (c_2+c_3) e^{13 t}-3 c_3\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{5 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-5 t} \left (c_1 \left (e^{10 t}+2 e^{13 t}-3\right )+(c_3-2 c_2) e^{10 t}+2 (c_2+c_3) e^{13 t}+3 c_3\right ) \\ \end{align*}