76.10.1 problem 1

Internal problem ID [17452]
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 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 1
Date solved : Monday, March 31, 2025 at 04:14:06 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 2 \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 19

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

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 22

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

\begin{align*} x(t)\to 4 e^{-t} \\ y(t)\to 2 e^{-2 t} \\ \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 15

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

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