Final answer:
The equilibrium solutions for the differential equation are y = √2 and y = -√2, found by setting dy/dt to zero and solving for y.
Step-by-step explanation:
To find the equilibrium solutions of the differential equation dy/dt = ((t² - 1)(y² - 2)) / (y² - 4), we need to set dy/dt to zero and solve for y, since equilibrium solutions occur when the rate of change (dy/dt) is zero. The denominator (y² - 4) does not affect the equilibrium since it does not involve t, so we focus on the numerator (t² - 1)(y² - 2). Setting each factor separately to zero gives us the potential equilibrium solutions as:
- t² - 1 = 0 results in t = ±1, but this does not give a value for y, hence it is not an equilibrium solution for y.
- y² - 2 = 0 results in y = ±√2, and these are the equilibrium solutions for y.
Therefore, the equilibrium solutions for the variable y in the given differential equation are y = √2 and y = -√2.