83.13.5 problem 9

Internal problem ID [22013]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IX. System of equations. Ex. XVII at page 154
Problem number : 9
Date solved : Thursday, October 02, 2025 at 08:21:40 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 26

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

\begin{align*} x \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= -c_2 \,{\mathrm e}^{-4 t}+3 c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

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

\begin{align*} x(t)&\to \frac {1}{4} e^{-4 t} \left (c_1 \left (e^{4 t}+3\right )+c_2 \left (e^{4 t}-1\right )\right )\\ y(t)&\to \frac {1}{4} e^{-4 t} \left (3 c_1 \left (e^{4 t}-1\right )+c_2 \left (3 e^{4 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.045 (sec). Leaf size: 22

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

\[ \left [ x{\left (t \right )} = \frac {C_{1}}{3} - C_{2} e^{- 4 t}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- 4 t}\right ] \]