69.29.3 problem 804

Internal problem ID [18544]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 804
Date solved : Thursday, October 02, 2025 at 03:14:44 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 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 )&=0 \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 9

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 10

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

\begin{align*} x(t)&\to 0\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*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)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (5 - \sqrt {29}\right ) e^{- \frac {t \left (1 + \sqrt {29}\right )}{2}}}{2} + \frac {C_{2} \left (5 + \sqrt {29}\right ) e^{- \frac {t \left (1 - \sqrt {29}\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (1 + \sqrt {29}\right )}{2}} + C_{2} e^{- \frac {t \left (1 - \sqrt {29}\right )}{2}}\right ] \]