76.8.9 problem 9

Internal problem ID [17427]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 9
Date solved : Monday, March 31, 2025 at 04:13:30 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.156 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 34

ode={D[x[t],t]==x[t]-5*y[t],D[y[t],t]==x[t]-3*y[t]}; 
ic={x[0]==1,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} (\cos (t)-3 \sin (t)) \\ y(t)\to e^{-t} (\cos (t)-\sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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