# Find the type of equilibrium at (0,0) for x'=x, y'=y

• anicic
In summary, the conversation discusses solving differential equations and finding the type and stability of an equilibrium point without using matrices. The equations given are x' = x and y' = y, and the solution involves integrating and rearranging them to get ln(y/x) = c. The conversation ends with a question about the behavior of the derivatives when x and y are close to 0 but positive.
anicic

## Homework Statement

I need to solve the differential equations given, find the type and stability of the equilibrium at (0,0), without matricies.

none

## The Attempt at a Solution

starting:

x' = x
y' = y

dy/dx = y'/x' = y/x

=> dy/y = dx/x

integrating gives

ln y + c1 = lnx + c2

rearrange and get

ln(y/x) = c

have i done it right so far?
i get stuck at this point

I'm only wondering why you are solving for y as a function of x at all.

dx/dt= x, dy/dt= y. What is true about the derivatives if x and y both close to 0 but positive?

## 1. What does "x'=x" mean in the equation?

The notation "x'=x" means that the rate of change of the variable x is equal to the current value of x. This is also known as a first-order autonomous differential equation.

## 2. What is an equilibrium point?

An equilibrium point is a point in a system where the values of all variables remain constant over time. In other words, the rates of change for all variables are equal to zero at this point.

## 3. How do you determine the type of equilibrium at (0,0)?

To determine the type of equilibrium at (0,0), we need to look at the behavior of the system near this point. If the system converges to (0,0) for all initial conditions, then it is a stable equilibrium. If the system diverges away from (0,0), then it is an unstable equilibrium. If the system neither converges nor diverges, it is a semi-stable equilibrium.

## 4. What is the significance of an equilibrium point in a system?

Equilibrium points are important in understanding the behavior of a system. They represent the state of the system where there is no net change, and can help predict the long-term behavior of the system. They can also indicate stability or instability in the system.

## 5. Can the type of equilibrium at (0,0) change?

Yes, the type of equilibrium at (0,0) can change depending on the values of the variables and the parameters in the system. As the system parameters change, the equilibrium point may shift or the type of equilibrium may change from stable to unstable or vice versa.

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