9.4.13 problem problem 13

Internal problem ID [977]
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 13
Date solved : Tuesday, March 04, 2025 at 12:06:43 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = 5*x__1(t)-9*x__2(t), diff(x__2(t),t) = 2*x__1(t)-x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 66

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

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+(c_1-3 c_2) \sin (3 t)) \\ \text {x2}(t)\to \frac {1}{3} e^{2 t} (3 c_2 \cos (3 t)+(2 c_1-3 c_2) \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 68

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

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