76.8.17 problem 17

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 : 17
Date solved : Thursday, March 13, 2025 at 10:08:11 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.099 (sec). Leaf size: 71

ode:=[diff(x(t),t) = -x(t)+a*y(t), diff(y(t),t) = -x(t)-y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-1+\sqrt {-a}\right ) t}+c_{2} {\mathrm e}^{-t \sqrt {-a}-t} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{\left (-1+\sqrt {-a}\right ) t}-c_{2} {\mathrm e}^{-t \sqrt {-a}-t}}{\sqrt {-a}} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 144

ode={D[x[t],t]==-x[t]+a*y[t],D[y[t],t]==-x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {e^{-\left (\left (\sqrt {-a}+1\right ) t\right )} \left (\sqrt {-a} c_1 \left (e^{2 \sqrt {-a} t}+1\right )+a c_2 \left (e^{2 \sqrt {-a} t}-1\right )\right )}{2 \sqrt {-a}} \\ y(t)\to \frac {e^{-\left (\left (\sqrt {-a}+1\right ) t\right )} \left (\sqrt {-a} c_2 \left (e^{2 \sqrt {-a} t}+1\right )-c_1 \left (e^{2 \sqrt {-a} t}-1\right )\right )}{2 \sqrt {-a}} \\ \end{align*}

✓ Sympy. Time used: 0.168 (sec). Leaf size: 71

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

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