70.7.12 problem 12

Internal problem ID [18772]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 12
Date solved : Thursday, October 02, 2025 at 03:30:44 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 48

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

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

✓ Sympy. Time used: 0.035 (sec). Leaf size: 22

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

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