14.22.3 problem 3

Internal problem ID [2730]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:12:25 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 66

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

\begin{align*} x_{1} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}+2 c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-t} c_1 \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}-\frac {5 c_3 \,{\mathrm e}^{-t}}{2}-{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 135

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

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (4 e^{9 t}+5\right )+2 (c_2+2 c_3) \left (e^{9 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (2 c_1 \left (e^{9 t}-1\right )+c_2 \left (e^{9 t}+8\right )+2 c_3 \left (e^{9 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (4 c_1 \left (e^{9 t}-1\right )+2 c_2 \left (e^{9 t}-1\right )+c_3 \left (4 e^{9 t}+5\right )\right ) \\ \end{align*}