87.20.14 problem 15

Internal problem ID [23699]
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 : 15
Date solved : Thursday, October 02, 2025 at 09:44:23 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*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 23

ode:=[diff(x(t),t) = 5*x(t)-6*y(t)+1, diff(y(t),t) = 6*x(t)-7*y(t)+1]; 
ic:=[x(0) = 0, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 26

ode={D[x[t],t]==5*x[t]-6*y[t]+1,D[y[t],t]==6*x[t]-7*y[t]+1}; 
ic={x[0]==0,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 1-e^{-t}\\ y(t)&\to 1-e^{-t} \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 17

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 = {x(0): 0, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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