Question

The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation dh/dt=(-R/v)(h/(k+h)) where is the hormone concentration in the bloodstream, is time, R is the maximum transport rate, V is the volume of the capillary, and is a positive constant that measures the affinity between the hormones and the enzymes that assist the process. Solve this differential equation to find a relationship between h and t.

Answer 1

`k~\text{log}h + h = (-R(t+c))/(v)` is the solution to the given differential equation.

Explanation:

We are given the following information in the graph:

`\displaystyle(dh)/(dt) = \bigg((-R)/(v)\bigg)\bigg((h)/(k+h)\bigg)`

where h is the hormone concentration in the bloodstream, R is the maximum transport rate, t is time, v is the volume of the capillary, and k is a positive constant that measures the affinity between the hormones.

We have to solve the given differential equation:

`\displaystyle(dh)/(dt) = \bigg((-R)/(v)\bigg)\bigg((h)/(k+h)\bigg)\\\\\bigg((-v)/(R)\bigg)\bigg((k+h)/(h)\bigg) dh = dt\\\\\bigg((-v)/(R)\bigg)\bigg((k)/(h) + 1\bigg) dh = dt\\\\\text{Integrating both sides}\\\\\int \bigg((-v)/(R)\bigg)\bigg((k)/(h) + 1\bigg) dh = \int dt\\\\\bigg((-v)/(R)\bigg)\bigg(k~\text{log}h + h\bigg) = t + c\\\\\text{where c is the constant of integration}\\\\k~\text{log}h + h = (-R(t+c))/(v)`