Final answer:
To find the equilibria and sketch the phase line for the equation u' = u²(3 - u), we need to find the equilibrium points by setting u' = 0 and solve for u. The equation has three solutions: u = 0, u = 3, and u = -∞. The equilibria are stable for u = 0 and u = 3, and unstable for u = -∞.
Step-by-step explanation:
In order to find the equilibria and sketch the phase line for the equation u' = u²(3 - u), we need to first find the equilibrium points by setting u' = 0 and solving for u. Setting u' = 0, we get u²(3 - u) = 0. This equation has three solutions: u = 0, u = 3, and u = -∞.
Now, we can determine the type and stability of the equilibria by analyzing the signs of u'. For u = 0, u' = 0, which means it is a stable equilibrium point. For u = 3, u' = 0, which means it is also a stable equilibrium point. Finally, for u = -∞, u' = 0, which means it is an unstable equilibrium point.
The phase line can be sketched by plotting the equilibrium points and determining the direction of the arrows based on the sign of u'. For u < 0, u' < 0, so the arrows point to the left. For 0 < u < 3, u' > 0, so the arrows point to the right. And for u > 3, u' < 0, so the arrows point to the left.
The phase line should have arrows pointing to the left for u < 0, arrows pointing to the right for 0 < u < 3, and arrows pointing to the left for u > 3.