9.6.26 problem problem 26

Internal problem ID [1033]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 26
Date solved : Monday, January 27, 2025 at 03:23:00 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 61

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (2 c_3 t +c_2 \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_3 \,t^{2}+c_2 t -4 c_3 t +c_1 -2 c_2 +2 c_3 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 105

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

\begin{align*} \text {x1}(t)\to e^{3 t} (2 c_1 t-c_2 t+c_3 t+c_1) \\ \text {x2}(t)\to \frac {1}{2} e^{3 t} \left ((2 c_1-c_2+c_3) t^2+2 c_1 t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{3 t} \left (c_3 \left (t^2-4 t+2\right )+2 c_1 (t-3) t-c_2 (t-4) t\right ) \\ \end{align*}