Which of these equations have an equilibrium solution. What are they? How are their DE's different?
dy/dx=6y-y^2
dy/dx 15-3y
To determine which of these equations have equilibrium solutions, we need to set the derivatives equal to zero and solve for y.
1. dy/dx = 6y - y^2
Setting dy/dx equal to zero:
0 = 6y - y^2
Rearranging the equation:
y^2 - 6y = 0
Factoring out y:
y(y - 6) = 0
We can see that there are two possible equilibrium solutions: y = 0 and y = 6.
2. dy/dx = 15 - 3y
Setting dy/dx equal to zero:
0 = 15 - 3y
Rearranging the equation:
3y = 15
Dividing by 3:
y = 5
We find that the only equilibrium solution for this equation is y = 5.
The differential equations (DE) for the two equations are different. The first equation, dy/dx = 6y - y^2, is a non-linear differential equation because of the presence of the y^2 term. The second equation, dy/dx = 15 - 3y, is a linear differential equation because the y term is not squared or multiplied by any other function of y.