9.6.5 problem problem 5

Internal problem ID [1012]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 5
Date solved : Saturday, March 29, 2025 at 10:36:50 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=7 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 7*x__1(t)+x__2(t), diff(x__2(t),t) = -4*x__1(t)+3*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 45

ode={D[ x1[t],t]==7*x1[t]+1*x2[t],D[ x2[t],t]==-4*x1[t]+3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{5 t} (2 c_1 t+c_2 t+c_1) \\ \text {x2}(t)\to e^{5 t} (c_2-2 (2 c_1+c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-7*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(4*x__1(t) - 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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