10.16.3 problem 3

Internal problem ID [1403]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 3
Date solved : Saturday, March 29, 2025 at 10:54:19 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 37

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= -\frac {c_1 \cos \left (t \right )}{5}+\frac {c_2 \sin \left (t \right )}{5}+\frac {2 c_1 \sin \left (t \right )}{5}+\frac {2 c_2 \cos \left (t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 41

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

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

✓ Sympy. Time used: 0.087 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) + 5*x__2(t) + Derivative(x__1(t), t),0),Eq(-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)
 

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