87.30.8 problem 8

Internal problem ID [23907]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 310
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:46:26 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )-6 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=6 x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 5*x(t)-6*y(t), diff(y(t),t) = 6*x(t)-7*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (6 c_2 t +6 c_1 -c_2 \right )}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 46

ode={D[x[t],t]==5*x[t]-6*y[t],D[y[t],t]==6*x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} (6 c_1 t-6 c_2 t+c_1)\\ y(t)&\to e^{-t} (6 (c_1-c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 6*y(t) + Derivative(x(t), t),0),Eq(-6*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 6 C_{1} t e^{- t} + \left (C_{1} + 6 C_{2}\right ) e^{- t}, \ y{\left (t \right )} = 6 C_{1} t e^{- t} + 6 C_{2} e^{- t}\right ] \]