68.1.61 problem 83

Internal problem ID [17128]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 83
Date solved : Thursday, October 02, 2025 at 01:44:08 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 33

ode:=[diff(x(t),t) = -5*x(t)+4*y(t), diff(y(t),t) = 2*x(t)+2*y(t)]; 
ic:=[x(0) = 4, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {32 \,{\mathrm e}^{-6 t}}{9}+\frac {4 \,{\mathrm e}^{3 t}}{9} \\ y \left (t \right ) &= -\frac {8 \,{\mathrm e}^{-6 t}}{9}+\frac {8 \,{\mathrm e}^{3 t}}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

ode={D[x[t],t]==-5*x[t]+4*y[t],D[y[t],t]==2*x[t]+2*y[t]}; 
ic={x[0]==4,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {4}{9} e^{-6 t} \left (e^{9 t}+8\right )\\ y(t)&\to \frac {8}{9} e^{-6 t} \left (e^{9 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 4 C_{1} e^{- 6 t} + \frac {C_{2} e^{3 t}}{2}, \ y{\left (t \right )} = C_{1} e^{- 6 t} + C_{2} e^{3 t}\right ] \]