9.5.1 problem Example 1

Internal problem ID [1004]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Examples. Page 437
Problem number : Example 1
Date solved : Tuesday, March 04, 2025 at 12:07:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+4 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 57

ode:=[diff(x__1(t),t) = 9*x__1(t)+4*x__2(t), diff(x__2(t),t) = -6*x__1(t)-x__2(t), diff(x__3(t),t) = 6*x__1(t)+4*x__2(t)+3*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{5 t}-\frac {3 c_3 \,{\mathrm e}^{3 t}}{2} \\ x_{3} \left (t \right ) &= c_2 \,{\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t}+c_1 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 113

ode={D[ x1[t],t]==9*x1[t]+4*x2[t]+0*x3[t],D[ x2[t],t]==-6*x1[t]-1*x2[t]+0*x3[t],D[ x3[t],t]==6*x1[t]+4*x2[t]+3*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} \left (c_1 \left (3 e^{2 t}-2\right )+2 c_2 \left (e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to -e^{3 t} \left (3 c_1 \left (e^{2 t}-1\right )+c_2 \left (2 e^{2 t}-3\right )\right ) \\ \text {x3}(t)\to \int _1^t3 x(K[1])dK[1]+\frac {6}{5} c_1 \left (e^{5 t}-1\right )+\frac {4}{5} c_2 \left (e^{5 t}-1\right )+c_3 \\ \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-9*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(6*x__1(t) + x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) - 4*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {2 C_{1} e^{3 t}}{3} + C_{2} e^{5 t}, \ x^{2}{\left (t \right )} = C_{1} e^{3 t} - C_{2} e^{5 t}, \ x^{3}{\left (t \right )} = C_{2} e^{5 t} + C_{3} e^{3 t}\right ] \]