9.6.16 problem problem 16

Internal problem ID [1023]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 16
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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-2 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+3 x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 38

dsolve([diff(x__1(t),t)=1*x__1(t)+0*x__2(t)-0*x__3(t),diff(x__2(t),t)=-2*x__1(t)-2*x__2(t)-3*x__3(t),diff(x__3(t),t)=2*x__1(t)+3*x__2(t)+4*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 (3 c_2 t +3 c_1 +c_2 +2 c_3 \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 57

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