9.6.12 problem problem 12

Internal problem ID [1019]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 12
Date solved : Monday, January 27, 2025 at 03:22:57 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 61

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 89

DSolve[{D[ x1[t],t]==-1*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==0*x1[t]-1*x2[t]+1*x3[t],D[ x3[t],t]==1*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}{2} e^{-t} \left (c_1 \left (t^2+2\right )+t (2 c_3-c_2 t)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2) t^2+2 c_3 t+2 c_2\right ) \\ \text {x3}(t)\to e^{-t} ((c_1-c_2) t+c_3) \\ \end{align*}