72.15.8 problem 19 (v)

Internal problem ID [14810]
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 : 19 (v)
Date solved : Thursday, March 13, 2025 at 04:19:23 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 27

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

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

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

ode={D[x[t],t]==2*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 e^{2 t} \\ y(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+3 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.081 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 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)
 

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