Depending on the type of soil there is a constant M that represents the maximum amount of water the soil can absorb per cubic ft. If the rate of absorption is proportional to the difference between the maximum amount of water that could be absorbed and the amount of water that has been absorbed, write a differential equation satisfied by y = f(t), the amount of water in the soil at time t. Does this equation accurately describe this situation: $\mathrm { y } ^ { \prime } = \mathrm { k } ( \mathrm { M } - \mathrm { y } ) ; \mathrm { k } 0 ?$

Question

Depending on the type of soil there is a constant M that represents the maximum amount of water the soil can absorb per cubic ft. If the rate of absorption is proportional to the difference between the maximum amount of water that could be absorbed and the amount of water that has been absorbed, write a differential equation satisfied by y = f(t), the amount of water in the soil at time t. Does this equation accurately describe this situation: $\mathrm { y } ^ { \prime } = \mathrm { k } ( \mathrm { M } - \mathrm { y } ) ; \mathrm { k } > 0 ?$

Quizplus · Accepted Answer

The given differential equation $y' = k(M - y)$ accurately describes the situation where the rate of absorption of water into the soil is proportional to the difference between the maximum capacity of water the soil can hold (M) and the current amount of water in the soil (y). This is a common form of a first-order linear differential equation representing processes where the rate of change of a quantity is proportional to the difference between a maximum value and the current value.

Depending on the Type of Soil There Is a Constant y′=k(M−y);k>0?\mathrm { y } ^ { \prime } = \mathrm { k } ( \mathrm { M } - \mathrm { y } ) ; \mathrm { k } > 0 ?y′=k(M−y);k>0?

Depending on the Type of Soil There Is a Constant $\mathrm { y } ^ { \prime } = \mathrm { k } ( \mathrm { M } - \mathrm { y } ) ; \mathrm { k } > 0 ?$