10.17.5 problem 5

Internal problem ID [1420]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:57:03 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 )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 75

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

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 164

DSolve[{D[ x1[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[ x2[t],t]==2*x1[t]+1*x2[t]-1*x3[t],D[ x3[t],t]==0*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}{3} e^{-t} \left (c_1 \left (2 e^{3 t}+1\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (6 t+4)-4\right )+c_2 \left (e^{3 t} (3 t+5)+4\right )+c_3 \left (e^{3 t} (3 t-4)+4\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (2-6 t)-2\right )+c_2 \left (2-e^{3 t} (3 t+2)\right )-c_3 \left (e^{3 t} (3 t-7)-2\right )\right ) \\ \end{align*}