83.13.6 problem 10

Internal problem ID [22014]
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 : 10
Date solved : Thursday, October 02, 2025 at 08:21:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+c_2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {3 c_1 \,{\mathrm e}^{4 t}}{2}-c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 0.052 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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