20.17.7 problem 7

Internal problem ID [3861]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 7
Date solved : Monday, January 27, 2025 at 08:03:33 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=15 x_{1} \left (t \right )-32 x_{2} \left (t \right )+12 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-17 x_{2} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=15*x__1(t)-32*x__2(t)+12*x__3(t),diff(x__2(t),t)=8*x__1(t)-17*x__2(t)+6*x__3(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)-1*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (16 c_{2} t +12 c_3 +16 c_{1} -c_{2} \right )}{32} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

DSolve[{D[x1[t],t]==15*x1[t]-32*x2[t]+12*x3[t],D[x2[t],t]==8*x1[t]-17*x2[t]+6*x3[t],D[x3[t],t]==0*x1[t]+0*x2[t]-1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} (16 c_1 t-32 c_2 t+12 c_3 t+c_1) \\ \text {x2}(t)\to e^{-t} (2 (4 c_1-8 c_2+3 c_3) t+c_2) \\ \text {x3}(t)\to c_3 e^{-t} \\ \end{align*}