67.5.16 problem Problem 3(e)

Internal problem ID [14031]
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(e)
Date solved : Monday, March 31, 2025 at 08:22:28 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-3 x \left (t \right )+2 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )+3 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 70

ode:=[diff(x(t),t)-3*x(t)+2*y(t) = 0, diff(y(t),t)-x(t)+3*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\sqrt {7}\, t}+c_2 \,{\mathrm e}^{-\sqrt {7}\, t} \\ y \left (t \right ) &= -\frac {c_1 \sqrt {7}\, {\mathrm e}^{\sqrt {7}\, t}}{2}+\frac {c_2 \sqrt {7}\, {\mathrm e}^{-\sqrt {7}\, t}}{2}+\frac {3 c_1 \,{\mathrm e}^{\sqrt {7}\, t}}{2}+\frac {3 c_2 \,{\mathrm e}^{-\sqrt {7}\, t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 144

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

\begin{align*} x(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (c_1 \left (\left (7+3 \sqrt {7}\right ) e^{2 \sqrt {7} t}+7-3 \sqrt {7}\right )-2 \sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right ) \\ y(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (\sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )-c_2 \left (\left (3 \sqrt {7}-7\right ) e^{2 \sqrt {7} t}-7-3 \sqrt {7}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 2*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)
 

\[ \left [ x{\left (t \right )} = C_{1} \left (3 - \sqrt {7}\right ) e^{- \sqrt {7} t} + C_{2} \left (\sqrt {7} + 3\right ) e^{\sqrt {7} t}, \ y{\left (t \right )} = C_{1} e^{- \sqrt {7} t} + C_{2} e^{\sqrt {7} t}\right ] \]