72.9.11 problem 25

Internal problem ID [14728]
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 : 25
Date solved : Thursday, March 13, 2025 at 04:17:48 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

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

\begin{align*} x(t)\to -2 e^{2 t} t \\ y(t)\to 2 e^{2 t} (t+1) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 37

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

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