72.9.10 problem 24

Internal problem ID [14727]
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 3.1. page 258
Problem number : 24
Date solved : Thursday, March 13, 2025 at 04:17:47 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = -2\\ y \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

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

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

✓ Sympy. Time used: 0.088 (sec). Leaf size: 29

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

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