Question

Let d y/d x=(y-1)(y-2)² y²

Find all critical points and sketch the phase diagram (or line) given below

Answer 1

Critical points of the differential equation dy/dx = (y-1)(y-2)²y² are y=0, y=1, and y=2. A phase diagram can be created by plotting these points and analyzing the sign of dy/dx around them. At y=1 and y=2, we have stable equilibria, and at y=0 we have an unstable equilibrium.

Step-by-step explanation:

The given differential equation is dy/dx = (y-1)(y-2)²y². To find the critical points of the equation, we set the differential equation equal to zero and solve for y. The critical points occur when the derivative is zero or undefined. In this case:

• If y = 1, then dy/dx = 0.
• If y = 2, then dy/dx = 0 since (y-2)² will be zero.
• If y = 0, then dy/dx = 0 since y² will be zero.

Thus, the critical points are at y = 0, 1, and 2. To sketch the phase diagram, plot these points on the y-axis and determine the behavior of the solution curves around these points. Solutions will increase where dy/dx > 0 and will decrease where dy/dx < 0. At y=1 and y=2, we have stable equilibria as the function approaches these values from either side, the slope decreases to 0. At y=0, we have an unstable equilibrium because as y moves away from 0, the slope increases.