87.19.4 problem 4

Internal problem ID [23674]
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 : 4
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 )+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: 0.050 (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: 0.428 (sec). Leaf size: 122

ode={D[x[t],t]==5*x[t]-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 \frac {1}{348} e^{-\left (\left (1+\sqrt {30}\right ) t\right )} \left (\left (36+7 \sqrt {30}\right ) e^{2 \sqrt {30} t}-72 e^{\sqrt {30} t+t}+36-7 \sqrt {30}\right )\\ y(t)&\to \frac {1}{58} e^{-\left (\left (1+\sqrt {30}\right ) t\right )} \left (\left (1+\sqrt {30}\right ) e^{2 \sqrt {30} t}-2 e^{\sqrt {30} t+t}+1-\sqrt {30}\right ) \end{align*}

✓ Sympy. Time used: 0.458 (sec). Leaf size: 88

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

\[ \left [ x{\left (t \right )} = - \frac {6}{29} + \frac {\left (36 - 7 \sqrt {30}\right ) e^{- t \left (1 + \sqrt {30}\right )}}{348} + \frac {\left (36 + 7 \sqrt {30}\right ) e^{- t \left (1 - \sqrt {30}\right )}}{348}, \ y{\left (t \right )} = - \frac {1}{29} + \frac {\left (1 - \sqrt {30}\right ) e^{- t \left (1 + \sqrt {30}\right )}}{58} + \frac {\left (1 + \sqrt {30}\right ) e^{- t \left (1 - \sqrt {30}\right )}}{58}\right ] \]