9.4.6 problem problem 6

Internal problem ID [970]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 6
Date solved : Tuesday, March 04, 2025 at 12:06:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )+5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = 9*x__1(t)+5*x__2(t), diff(x__2(t),t) = -6*x__1(t)-2*x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 6 \,{\mathrm e}^{4 t}-5 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -6 \,{\mathrm e}^{4 t}+6 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 33

ode={D[ x1[t],t]==9*x1[t]+5*x2[t],D[ x2[t],t]==-6*x1[t]-2*x2[t]}; 
ic={x1[0]==1,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} \left (6 e^t-5\right ) \\ \text {x2}(t)\to -6 e^{3 t} \left (e^t-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-9*x__1(t) - 5*x__2(t) + Derivative(x__1(t), t),0),Eq(6*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {5 C_{1} e^{3 t}}{6} - C_{2} e^{4 t}, \ x^{2}{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{4 t}\right ] \]