14.22.5 problem 5

Internal problem ID [2732]
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 : 5
Date solved : Monday, January 27, 2025 at 06:12:27 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-7 x_{1} \left (t \right )+6 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=5 x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 44

dsolve([diff(x__1(t),t)=-7*x__1(t)+0*x__2(t)+6*x__3(t),diff(x__2(t),t)=0*x__1(t)+5*x__2(t)+0*x__3(t),diff(x__3(t),t)=6*x__1(t)+0*x__2(t)+2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{5 t}+c_2 \,{\mathrm e}^{-10 t} \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{5 t} \\ x_{3} \left (t \right ) &= 2 c_1 \,{\mathrm e}^{5 t}-\frac {c_2 \,{\mathrm e}^{-10 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 158

DSolve[{D[ x1[t],t]==-7*x1[t]+0*x2[t]+6*x3[t],D[ x2[t],t]==0*x1[t]+5*x2[t]+0*x3[t],D[ x3[t],t]==6*x1[t]+0*x2[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-10 t} \left (c_1 \left (e^{15 t}+4\right )+2 c_2 \left (e^{15 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-10 t} \left (2 c_1 \left (e^{15 t}-1\right )+c_2 \left (4 e^{15 t}+1\right )\right ) \\ \text {x2}(t)\to c_3 e^{5 t} \\ \text {x1}(t)\to \frac {1}{5} e^{-10 t} \left (c_1 \left (e^{15 t}+4\right )+2 c_2 \left (e^{15 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-10 t} \left (2 c_1 \left (e^{15 t}-1\right )+c_2 \left (4 e^{15 t}+1\right )\right ) \\ \text {x2}(t)\to 0 \\ \end{align*}