Find the particular solution of this differential equation that satisfies the given initial condition.

In the following answer, I assume that k and are constants.

Then, the given differential equation can be solved by separation of variables:

Dividing by P results in

.

Integrating both sides, we obtain

, where C is an arbitrary constant. We can now solve for P(t) by rewriting the natural logarithmic equation as an exponential (with the base e) equation:

.

So, the general solution of the equation is . Since the initial condition is , we can find C:

. Therefore,

The particular solution of the equation with the given initial condition is