44.1.47 problem 49

Internal problem ID [6922]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 49
Date solved : Wednesday, March 05, 2025 at 02:50:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 35

ode:=[diff(x(t),t) = x(t)+3*y(t), diff(y(t),t) = 5*x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{6 t} c_{1} +{\mathrm e}^{-2 t} c_{2} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{6 t} c_{1}}{3}-{\mathrm e}^{-2 t} c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode={D[x[t],t]==x[t]+3*y[t],D[y[t],t]==5*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{8} e^{-2 t} \left (c_1 \left (3 e^{8 t}+5\right )+3 c_2 \left (e^{8 t}-1\right )\right ) \\ y(t)\to \frac {1}{8} e^{-2 t} \left (5 c_1 \left (e^{8 t}-1\right )+c_2 \left (5 e^{8 t}+3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 2 t} + \frac {3 C_{2} e^{6 t}}{5}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{6 t}\right ] \]