# Differential equation- find max slope

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In summary, the Gompertz function is a solution of the differential equation dP/dt = c*ln(K/P)*P, where c is a constant and K is the carrying capacity. To find the value of P at which P grows fastest, we can set the second derivative of dP/dt to 0 and solve for t. Plugging this value of t back into the original function will give us the maximum value of P.
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## Homework Statement

Another model for a growth function for a limited pupulation is given by the Gompertz function, which is a solution of the differential equation

dP/dt = c*ln(K/P)*P

where 'c' is a constant and 'K' is the carrying capacity.

At what value of P does P grow fastest?

## Homework Equations

c = .05
K=1000
P_0 = 500 (initial condition)

P(t) = 1000/e^(e^(-.05t-.3665)) (this is the specific solution)

## The Attempt at a Solution

I think it is asking to find what the max value of dP/dt. However I think to do this, I need to find the derivative of dP/dt which is d^2P/dt^2 and set it equal to zero. This will let me find t when the slope of P is at its max and then I plug t back into P. Is this correct?

If it is, I am in for one hell of a derivative...

Thanks!

Why solve for P? You were asked for a value of P for which P grows fastest which mean its derivative is a maximum. You can find a maximum of a function by setting its derivative to 0. Here, the function is dP/dt so differentiate that:
$$\frac{d}{dt}\frac{dP}{dt}= \frac{d^2P}{dt^2}$$$$= C\frac{d(P(ln(K)- ln(P))}{dt}= 0$$

Ok. Solving that derivative I get t = -7.33 and in the end P has a max slope at P= 367.879. These are right.

Thanks!

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates the rates of change of a function to the function itself. It contains one or more derivatives of the unknown function.

## 2. How do you solve a differential equation?

The method for solving a differential equation depends on its type and order. It involves manipulating the equation to isolate the dependent variable and integrating both sides to find the general solution.

## 3. What is the purpose of finding the maximum slope in a differential equation?

The maximum slope, also known as the point of inflection, indicates the point where the rate of change of the function is the highest. It is useful in understanding the behavior and stability of the system described by the differential equation.

## 4. How do you find the maximum slope in a differential equation?

To find the maximum slope, we first need to solve the differential equation to obtain the function. Then, we can find the derivative of the function and set it equal to zero to find the critical points. The critical point with the highest value of the derivative will be the maximum slope.

## 5. Can the maximum slope be negative in a differential equation?

Yes, the maximum slope can be negative in a differential equation. This indicates that the function is decreasing at that point and the maximum slope is in the opposite direction of the function's growth.

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