72.12.14 problem 24

Internal problem ID [14782]
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 : 24
Date solved : Thursday, March 13, 2025 at 04:18:52 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 22

ode={D[x[t],t]==-3*x[t]-y[t],D[y[t],t]==4*x[t]+y[t]}; 
ic={x[0]==-1,y[0]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -e^{-t} \\ y(t)\to 2 e^{-t} \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 37

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

\[ \left [ x{\left (t \right )} = - 2 C_{2} t e^{- t} - \left (2 C_{1} - C_{2}\right ) e^{- t}, \ y{\left (t \right )} = 4 C_{1} e^{- t} + 4 C_{2} t e^{- t}\right ] \]