72.10.17 problem 13 (a)

Internal problem ID [14749]
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 : 13 (a)
Date solved : Thursday, March 13, 2025 at 04:18:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 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: 0.044 (sec). Leaf size: 33

ode:=[diff(x(t),t) = -4*x(t)+y(t), diff(y(t),t) = 2*x(t)-3*y(t)]; 
ic:=x(0) = 1y(0) = 0; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 40

ode={D[x[t],t]==-4*x[t]+y[t],D[y[t],t]==2*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 \frac {1}{3} e^{-5 t} \left (e^{3 t}+2\right ) \\ y(t)\to \frac {2}{3} e^{-5 t} \left (e^{3 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*x(t) - y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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