10.16.9 problem 9

Internal problem ID [1409]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 12:35:23 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = x__1(t)-5*x__2(t), diff(x__2(t),t) = x__1(t)-3*x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (-5 \cos \left (t \right )+5 \sin \left (t \right )\right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 34

ode={D[ x1[t],t]==1*x1[t]-5*x2[t],D[ x2[t],t]==1*x1[t]-3*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^{-t} (\cos (t)-3 \sin (t)) \\ \text {x2}(t)\to e^{-t} (\cos (t)-\sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 46

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

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