72.12.5 problem 5

Internal problem ID [14773]
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 3.5 page 327
Problem number : 5
Date solved : Thursday, March 13, 2025 at 04:18:42 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 17

ode:=[diff(x(t),t) = -3*x(t), diff(y(t),t) = x(t)-3*y(t)]; 
ic:=x(0) = 1y(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \\ y &= t \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 20

ode={D[x[t],t]==-3*x[t]+0*y[t],D[y[t],t]==1*x[t]-3*y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-3 t} \\ y(t)\to e^{-3 t} t \\ \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + Derivative(x(t), t),0),Eq(-x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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