72.10.5 problem 5

Internal problem ID [14737]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.2. page 277
Problem number : 5
Date solved : Thursday, March 13, 2025 at 04:17:58 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 23

ode:=[diff(x(t),t) = -1/2*x(t), diff(y(t),t) = x(t)-1/2*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 33

ode={D[x[t],t]==-1/2*x[t],D[y[t],t]==x[t]-1/2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{-t/2} \\ y(t)\to e^{-t/2} (c_1 t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.071 (sec). Leaf size: 26

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

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