78.5.17 problem 7.e

Internal problem ID [21056]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.e
Date solved : Thursday, October 02, 2025 at 07:01:49 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+2 y\\ y^{\prime }&=3 x \left (t \right )+3 y \end{align*}

✓ Maple. Time used: 0.101 (sec). Leaf size: 30

ode:=[diff(x(t),t) = 4*x(t)+2*y(t), diff(y(t),t) = 3*x(t)+3*y(t)]; 
dsolve(ode);
 

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

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

ode={D[x[t],t]==4*x[t]+2*y[t],D[y[t],t]==3*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} e^t \left (c_1 \left (3 e^{5 t}+2\right )+2 c_2 \left (e^{5 t}-1\right )\right )\\ y(t)&\to \frac {1}{5} e^t \left (3 c_1 \left (e^{5 t}-1\right )+c_2 \left (2 e^{5 t}+3\right )\right ) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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