10.16.1 problem 1

Internal problem ID [1401]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 1
Date solved : Saturday, March 29, 2025 at 10:54:16 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.129 (sec). Leaf size: 52

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t), diff(x__2(t),t) = 4*x__1(t)-x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 58

ode={D[ x1[t],t]==3*x1[t]-2*x2[t],D[ x2[t],t]==4*x1[t]-1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.110 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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