9.6.15 problem problem 15

Internal problem ID [1022]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 15
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 )&=-2 x_{1} \left (t \right )-9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=-2*x__1(t)-9*x__2(t)-0*x__3(t),diff(x__2(t),t)=1*x__1(t)+4*x__2(t)-0*x__3(t),diff(x__3(t),t)=1*x__1(t)+3*x__2(t)+1*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 62

DSolve[{D[ x1[t],t]==-2*x1[t]-9*x2[t]-0*x3[t],D[ x2[t],t]==1*x1[t]+4*x2[t]-0*x3[t],D[ x3[t],t]==1*x1[t]+3*x2[t]+1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -e^t (c_1 (3 t-1)+9 c_2 t) \\ \text {x2}(t)\to e^t ((c_1+3 c_2) t+c_2) \\ \text {x3}(t)\to e^t ((c_1+3 c_2) t+c_3) \\ \end{align*}