9.4.7 problem problem 7

Internal problem ID [971]
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 7
Date solved : Tuesday, March 04, 2025 at 12:06:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 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 )-5 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 30

ode:=[diff(x__1(t),t) = -3*x__1(t)+4*x__2(t), diff(x__2(t),t) = 6*x__1(t)-5*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} c_1 +c_2 \,{\mathrm e}^{-9 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} c_1 -\frac {3 c_2 \,{\mathrm e}^{-9 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode={D[ x1[t],t]==-3*x1[t]+4*x2[t],D[ x2[t],t]==6*x1[t]-5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-9 t} \left (c_1 \left (3 e^{10 t}+2\right )+2 c_2 \left (e^{10 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-9 t} \left (3 c_1 \left (e^{10 t}-1\right )+c_2 \left (2 e^{10 t}+3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) + 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {2 C_{1} e^{- 9 t}}{3} + C_{2} e^{t}, \ x^{2}{\left (t \right )} = C_{1} e^{- 9 t} + C_{2} e^{t}\right ] \]