72.15.2 problem 6

Internal problem ID [14804]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number : 6
Date solved : Thursday, March 13, 2025 at 04:19:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 27

ode={D[x[t],t]==0*x[t]+0*y[t],D[y[t],t]==1*x[t]-1*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.062 (sec). Leaf size: 12

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

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