87.20.5 problem 5

Internal problem ID [23690]
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 161
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:44:17 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=5 x-6 y \left (t \right )+1\\ y^{\prime }\left (t \right )&=6 x-7 y \left (t \right )+1 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 38

ode:=[diff(x(t),t) = 5*x(t)-6*y(t)+1, diff(y(t),t) = 6*x(t)-7*y(t)+1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 1+{\mathrm e}^{-t} \left (t c_1 +c_2 \right ) \\ y \left (t \right ) &= 1+\frac {{\mathrm e}^{-t} \left (6 t c_1 -c_1 +6 c_2 \right )}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode={D[x[t],t]==5*x[t]-6*y[t]+1,D[y[t],t]==6*x[t]-7*y[t]+1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.088 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 6*y(t) + Derivative(x(t), t) - 1,0),Eq(-6*x(t) + 7*y(t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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