Final answer:
To find the equilibrium solution to the differential equation dy/dt=0.5y-250, set dy/dt=0 and solve for y. The equilibrium solution is y=500. The general solution is ln|y|=0.5t+C, where C is the constant of integration.
Step-by-step explanation:
To find the equilibrium solution to the differential equation dy/dt=0.5y-250, we set dy/dt equal to zero and solve for y. So, 0.5y-250=0. From here, we can add 250 to both sides of the equation and then divide by 0.5 to solve for y. The equilibrium solution is y=500.
The general solution to this differential equation can be found by separating the variables and integrating. We start by moving the dy term to one side and the y-250 term to the other side. This gives us (1/y)dy=0.5dt. Next, we integrate both sides of the equation. On the left side, we integrate (1/y)dy to get ln|y|. On the right side, we integrate 0.5dt to get 0.5t. So, the general solution is ln|y|=0.5t+C, where C is the constant of integration.