72.10.16 problem 12 (c)

Internal problem ID [14748]
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 : 12 (c)
Date solved : Thursday, March 13, 2025 at 04:18:10 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 )-2 y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 26

ode:=[diff(x(t),t) = 3*x(t), diff(y(t),t) = x(t)-2*y(t)]; 
ic:=x(0) = 2y(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{3 t} \\ y &= \frac {2 \,{\mathrm e}^{3 t}}{5}+\frac {8 \,{\mathrm e}^{-2 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 31

ode={D[x[t],t]==3*x[t],D[y[t],t]==x[t]-2*y[t]}; 
ic={x[0]==2,y[0]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 2 e^{3 t} \\ y(t)\to \frac {2}{5} e^{-2 t} \left (e^{5 t}+4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.072 (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) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 5 C_{1} e^{3 t}, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{- 2 t}\right ] \]