9.6.7 problem problem 7

Internal problem ID [1014]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 7
Date solved : Monday, January 27, 2025 at 03:22:56 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-7 x_{1} \left (t \right )+9 x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 42

dsolve([diff(x__1(t),t)=2*x__1(t)+0*x__2(t)+0*x__3(t),diff(x__2(t),t)=-7*x__1(t)+9*x__2(t)+7*x__3(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{2 t}+{\mathrm e}^{2 t} c_3 +c_1 \,{\mathrm e}^{9 t} \\ x_{3} \left (t \right ) &= c_2 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 60

DSolve[{D[ x1[t],t]==2*x1[t]+0*x2[t]+0*x3[t],D[ x2[t],t]==-7*x1[t]+9*x2[t]+7*x3[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{2 t} \\ \text {x2}(t)\to e^{2 t} \left (-\left (c_1 \left (e^{7 t}-1\right )\right )+(c_2+c_3) e^{7 t}-c_3\right ) \\ \text {x3}(t)\to c_3 e^{2 t} \\ \end{align*}