10.16.10 problem 10

Internal problem ID [1410]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 10
Date solved : Saturday, March 29, 2025 at 10:54:30 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 )&=-x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2 \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = -3*x__1(t)+2*x__2(t), diff(x__2(t),t) = -x__1(t)-x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = -2; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (-5 \sin \left (t \right )+\cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (-6 \sin \left (t \right )-4 \cos \left (t \right )\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 27

ode={D[ x1[t],t]==-3*x1[t]+2*x2[t],D[ x2[t],t]==-1*x1[t]-1*x2[t]}; 
ic={x1[0]==1,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.102 (sec). Leaf size: 48

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(x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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