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 : Sunday, March 30, 2025 at 02:10:16 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*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 46

ode:=[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) = -x__3(t)]; 
dsolve(ode);
 

\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 +16 c_1 -c_2 +12 c_3 \right )}{32} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-t} \\ \end{align*}

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

ode={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]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-15*x__1(t) + 32*x__2(t) - 12*x__3(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) + 17*x__2(t) - 6*x__3(t) + Derivative(x__2(t), t),0),Eq(x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 16 C_{2} t e^{- t} + \left (- \frac {3 C_{1}}{4} + C_{2} + 16 C_{3}\right ) e^{- t}, \ x^{2}{\left (t \right )} = 8 C_{2} t e^{- t} + 8 C_{3} e^{- t}, \ x^{3}{\left (t \right )} = C_{1} e^{- t}\right ] \]