We are given this situation to model using a differential equation: Snow is falling at a constant rate of 3/5 in per hour and is being removed at a constant rate of 45% of the amount of snow on the ground per hour. Removal of snow began at 4 in. If we let h(t) be the height of the snow at time t, then the rate of change of the height of snow would be equal to 3/5 - 0.45h. My question is, why doesn't time appear in the differential equation? Our equation, dh/dt = 3/5 - 0.45h is autonomous, but why?
We are given this situation to model using a differential equation: Snow is falling at a constant rate of 3/5 in per hour and is being removed at a constant rate of 45% of the amount of snow on the ground per hour. Removal of snow began at 4 in. If we let h(t) be the height of the snow at time t, then the rate of change of the height of snow would be equal to 3/5 - 0.45h. My question is, why doesn't time appear in the differential equation? Our equation, dh/dt = 3/5 - 0.45h is autonomous, but why?
We are given this situation to model using a differential equation: Snow is falling at a constant rate of 3/5 in per hour and is being removed at a constant rate of 45% of the amount of snow on the ground per hour. Removal of snow began at 4 in. If we let h(t) be the height of the snow at time t, then the rate of change of the height of snow would be equal to 3/5 - 0.45h. My question is, why doesn't time appear in the differential equation? Our equation, dh/dt = 3/5 - 0.45h is autonomous, but why?
We are given this situation to model using a differential equation: Snow is falling at a constant rate of 3/5 in per hour and is being removed at a constant rate of 45% of the amount of snow on the ground per hour. Removal of snow began at 4 in. If we let h(t) be the height of the snow at time t, then the rate of change of the height of snow would be equal to 3/5 - 0.45h. My question is, why doesn't time appear in the differential equation? Our equation, dh/dt = 3/5 - 0.45h is autonomous, but why?
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Expert Solution
Step 1: Definition of an Autonomous Differential equation:
An autonomous differential equation is an equation of the form
Let's think of t as indicating time. This equation says that the rate of change of the function y(t) is given by a some rule. The rule says that if the current value is y then the rate of change is f(y).
The equation is called a differential equation, because it is an equation involving the derivative. .The differential equation is called autonomous because the rule doesn't care what time t it is.
