74.22.7 problem 7

Internal problem ID [16575]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 7
Date solved : Thursday, March 13, 2025 at 08:23:34 AM
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 )&=4 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 34

ode:=[diff(x(t),t) = -3*x(t)+6*y(t), diff(y(t),t) = 4*x(t)-y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-7 t} \\ y &= c_{1} {\mathrm e}^{3 t}-\frac {2 c_{2} {\mathrm e}^{-7 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 74

ode={D[x[t],t]==-3*x[t]+6*y[t],D[y[t],t]==4*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-7 t} \left (c_1 \left (2 e^{10 t}+3\right )+3 c_2 \left (e^{10 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-7 t} \left (2 c_1 \left (e^{10 t}-1\right )+c_2 \left (3 e^{10 t}+2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 34

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(-4*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {3 C_{1} e^{- 7 t}}{2} + C_{2} e^{3 t}, \ y{\left (t \right )} = C_{1} e^{- 7 t} + C_{2} e^{3 t}\right ] \]