The autonomous differential equation

with x(0) = 1. This describes a population with per capita production that decreases like 1/1 + x.

The following autonomous differential equations can be solved except for the step of finding x in terms of t. Use the steps to figure out how the solution behaves.

a. Solve the equation with separation of variables.

b. It is impossible to algebraically solve for x in terms of However, you can still find the arbitrary constant. Find it.

c. Although you cannot find x as a function of t, you can find t as a function of x. Graph this function for 1 ≤ x ≤ 10.

dxx di +