9.6.17 problem problem 17

Internal problem ID [1024]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 17
Date solved : Monday, January 27, 2025 at 03:22:58 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=18 x_{1} \left (t \right )+7 x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-27 x_{1} \left (t \right )-9 x_{2} \left (t \right )-5 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 40

dsolve([diff(x__1(t),t)=1*x__1(t)+0*x__2(t)-0*x__3(t),diff(x__2(t),t)=18*x__1(t)+7*x__2(t)+4*x__3(t),diff(x__3(t),t)=-27*x__1(t)-9*x__2(t)-5*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (c_2 t +c_1 \right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (6 c_2 t +6 c_1 -c_2 +18 c_3 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 63

DSolve[{D[ x1[t],t]==1*x1[t]+0*x2[t]-0*x3[t],D[ x2[t],t]==18*x1[t]+7*x2[t]+4*x3[t],D[ x3[t],t]==-27*x1[t]-9*x2[t]-5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to e^t (2 (9 c_1+3 c_2+2 c_3) t+c_2) \\ \text {x3}(t)\to e^t (c_3-3 (9 c_1+3 c_2+2 c_3) t) \\ \end{align*}