84.41.6 problem 31.7

Internal problem ID [22388]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 31. Solutions of linear systems with constant coefficients. Solved problems. Page 194
Problem number : 31.7
Date solved : Thursday, October 02, 2025 at 08:38:09 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+y\\ y^{\prime }&=9 x \left (t \right )+y \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 35

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = 9*x(t)+y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{4 t} \\ y \left (t \right ) &= -3 c_1 \,{\mathrm e}^{-2 t}+3 c_2 \,{\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

ode={D[x[t],t]==x[t]+y[t],D[y[t],t]==9*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{-2 t} \left (3 c_1 \left (e^{6 t}+1\right )+c_2 \left (e^{6 t}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{-2 t} \left (3 c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - y(t) + Derivative(x(t), t),0),Eq(-9*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{3} + \frac {C_{2} e^{4 t}}{3}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{4 t}\right ] \]