How can I numerically find a solution of a series of two differential equations.

So, I have to numerically find the solution to the following system (image attached) in a neighborhood of the equilibrium. Equilibrium in this case is x'(t)=0 & y'(t)=0.

Also these are the values of the constants. Alpha=4 Beta=3 L=2 k=1

What should be the MATLAB code for this? One friend suggested using ode45 but I do not know exactly how.

[In case you cannot view the image:

X’(t) = -alpha x(t) + k y(t)

Y’(t) = L x(t) – beta y(t)]

[Birdman]

There is no image attached.

[Muhammad Islam]

X’(t) = -alpha x(t) + k y(t)

Y’(t) = L x(t) – beta y(t)

Answer 1

[Roger Stafford]

This is hardly a matlab problem. At an equilibrium you would have the equations:

    -4*x + 1*y = 0
     2*x - 3*y = 0

The only possible simultaneous solution for that is x = y = 0.

The general solution to that problem is:

   x = C1*exp(-2*t)-C2*exp(-5*t)
   y = 2*C1*exp(-2*t)+C2*exp(-5*t)

where C1 and C2 are constants depending on initial conditions of x and y. To arrive at a "neighborhood" of x' = 0 and y' = 0, you would have to approach infinity for t. In other words, equilibrium is only approached asymptotically.