61.5.15 problem Problem 3(d)

Internal problem ID [15382]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(d)
Date solved : Thursday, October 02, 2025 at 10:12:24 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )+5 x \left (t \right )-2 y&=0\\ 2 x \left (t \right )+y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 82

ode:=[diff(x(t),t)+5*x(t)-2*y(t) = 0, diff(y(t),t)+2*x(t)-y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t}+c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \\ y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t} \sqrt {5}}{2}-\frac {c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \sqrt {5}}{2}+\frac {3 c_1 \,{\mathrm e}^{\left (-2+\sqrt {5}\right ) t}}{2}+\frac {3 c_2 \,{\mathrm e}^{-\left (2+\sqrt {5}\right ) t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 148

ode={D[x[t],t]+5*x[t]-2*y[t]==0,D[y[t],t]+2*x[t]-y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_1 \left (\left (5-3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5+3 \sqrt {5}\right )+2 \sqrt {5} c_2 \left (e^{2 \sqrt {5} t}-1\right )\right )\\ y(t)&\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_2 \left (\left (5+3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5-3 \sqrt {5}\right )-2 \sqrt {5} c_1 \left (e^{2 \sqrt {5} t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (3 - \sqrt {5}\right ) e^{- t \left (2 - \sqrt {5}\right )}}{2} + \frac {C_{2} \left (\sqrt {5} + 3\right ) e^{- t \left (2 + \sqrt {5}\right )}}{2}, \ y{\left (t \right )} = C_{1} e^{- t \left (2 - \sqrt {5}\right )} + C_{2} e^{- t \left (2 + \sqrt {5}\right )}\right ] \]