Question

Treat h as a constant. For what values of h (if any) does the following equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. dxdy​ =y³ −4y−h Which of the equilibria, if any, are stable? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. There is one stable equilibrium when, and there are no stable equilibria for other values of h. (Type your answer in inequality notation.)
B. There are two stable equilibria when, and there are no stable equilibria for other values of h. (Type your answer in inequality notation.)
C. All of the equilibria are stable for all values of h.
D. The equation does not have equilibria for any value of h.

Answer 1

The equilibria of the equation are the solutions to the cubic equation y³ - 4y = h. A graphical argument using a potential energy curve suggests that there can be at least one stable equilibrium, depending on h, as cubic equations typically have one or more minima (stable points). However, without a specific graph or more information, we cannot provide the exact values of h that result in stable equilibria.

Step-by-step explanation:

To determine the equilibria of the equation dy/dx = y³ − 4y − h, we set the right-hand side of the equation to zero and solve for y. This gives us the equilibria points as the real solutions of the cubic equation y³ − 4y = h. A graphical argument can be used to determine the stability of these equilibria by looking at the potential energy curve U(y) which is such that dU/dy = −(y³ − 4y), and where the equilibria correspond to points where the slope (dU/dy) is zero.

From the behavior of cubic equations and potential energy curves, we know that they typically have regions where the slope is positive and negative, representing stable and unstable equilibria, respectively. A stable equilibrium is where the curve has a minimum which would mean the second derivative of U with respect to y is positive (² U/dy² > 0), and that would correspond to values of h for which the curve crosses the y-axis at those minima. Without the precise graph or further information, we can't provide exact values of h for which there is one or more stable equilibria. However, we can assert that the cubic equation can have up to 3 real roots, with typically at least one minimum and one maximum, suggesting that there can be at least one stable equilibrium, depending on the value of h.