57.22.3 problem 4(c)

Internal problem ID [14514]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 4(c)
Date solved : Thursday, October 02, 2025 at 09:38:03 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=2 x+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=6 x+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 2*x(t)+2*y(t), diff(y(t),t) = 6*x(t)+3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 74

ode={D[x[t],t]==2*x[t]+2*y[t],D[y[t],t]==6*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{7} e^{-t} \left (c_1 \left (3 e^{7 t}+4\right )+2 c_2 \left (e^{7 t}-1\right )\right )\\ y(t)&\to \frac {1}{7} e^{-t} \left (6 c_1 \left (e^{7 t}-1\right )+c_2 \left (4 e^{7 t}+3\right )\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-6*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{- t}}{3} + \frac {C_{2} e^{6 t}}{2}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{6 t}\right ] \]