Suppose a cell of volume V cc is surrounded by a homogeneous chemical solution of concentration C g/cc. Let y denote the concentration of the solute inside the cell at any time t and suppose that, initially, the concentration is y0. Fick's law, named after the German physiologist Adolf Fick (1829-1901), states that the rate of change of the concentration of solute inside the cell at any time t is proportional to the difference between the concentration of the solute outside the cell and the concentration inside the cell and inversely proportional to the volume of the cell. Fick's law may be expressed as the differential equation.
dy/dt = k/V(C-Y)
y(0) = y_0
Use separation of variables to solve the differential equation given above, where k, V, C, and y0 are constants with
C − y > 0.
(Note: The constant of proportionality k depends on the area and permeability of the cell membrane.)
y = ?????
Hi Steve. Thanks for the reply! I'm having trouble getting started with it and would appreciate any help you could provide. Thanks!
To solve the given differential equation using separation of variables, we need to separate the variables and integrate both sides.
The differential equation is:
(dy/dt) = k/V(C - y)
Now, let's separate the variables:
(dy / (C - y)) = (k / V) dt
Next, we can integrate both sides:
∫ (dy / (C - y)) = ∫ (k / V) dt
To integrate the left side, we can use the method of partial fractions. Let's express the left side using partial fractions:
∫ (dy / (C - y)) = A / (C - y)
Multiplying both sides by (C - y), we get:
dy = A dt
Integrating both sides, we have:
∫ dy = ∫ A dt
y = At + B
Now, we need to determine the values of A and B. For that, we use the initial condition y(0) = y0:
y(0) = A(0) + B
y0 = B
Therefore, the solution to the differential equation is:
y = At + y0
This is the solution to the given differential equation using separation of variables.
This is just like the one you posted earlier at
https://www.jiskha.com/display.cgi?id=1507322985