72.10.12 problem 11 (b)

Internal problem ID [14752]
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 : 11 (b)
Date solved : Monday, March 31, 2025 at 12:57:27 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode={D[x[t],t]==-2*x[t]-2*y[t],D[y[t],t]==-2*x[t]+y[t]}; 
ic={x[0]==0,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {2}{5} e^{-3 t} \left (e^{5 t}-1\right ) \\ y(t)\to \frac {1}{5} e^{-3 t} \left (4 e^{5 t}+1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 34

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

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