87.19.5 problem 5

Internal problem ID [23675]
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 3. Linear Systems. Exercise at page 149
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:44:09 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.044 (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: 0.009 (sec). Leaf size: 145

ode={D[x[t],t]==5*x[t]-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 \frac {1}{60} e^{-\left (\left (1+\sqrt {30}\right ) t\right )} \left (6 c_1 \left (\left (5+\sqrt {30}\right ) e^{2 \sqrt {30} t}+5-\sqrt {30}\right )-\sqrt {30} c_2 \left (e^{2 \sqrt {30} t}-1\right )\right )\\ y(t)&\to \frac {1}{10} e^{-\left (\left (1+\sqrt {30}\right ) t\right )} \left (\sqrt {30} c_1 \left (e^{2 \sqrt {30} t}-1\right )-c_2 \left (\left (\sqrt {30}-5\right ) e^{2 \sqrt {30} t}-5-\sqrt {30}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 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)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (6 - \sqrt {30}\right ) e^{- t \left (1 + \sqrt {30}\right )}}{6} + \frac {C_{2} \left (\sqrt {30} + 6\right ) e^{- t \left (1 - \sqrt {30}\right )}}{6}, \ y{\left (t \right )} = C_{1} e^{- t \left (1 + \sqrt {30}\right )} + C_{2} e^{- t \left (1 - \sqrt {30}\right )}\right ] \]