Question Video: Finding the Solution of Logistic Differential Equations | Nagwa Question Video: Finding the Solution of Logistic Differential Equations | Nagwa

# Question Video: Finding the Solution of Logistic Differential Equations Mathematics • Higher Education

Suppose a populationβs growth is governed by the logistic equation dπ/dπ‘ = 0.07π(1 β (π/900)), where π(0) = 50. Write the formula for π(π‘).

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### Video Transcript

Suppose a populationβs growth is governed by the logistic equation dπ by dπ‘ equals 0.07π multiplied by one minus π over 900, where π of zero is equal to 50. Write the formula for π of π‘.

Writing π of π‘ means that we need to find the solution to this logistic equation. We can begin by writing down its general form. We know that for the logistic equation dπ by dπ‘ equals ππ multiplied by one minus π over πΏ that its solution is given by π equals πΏ over one plus π΄π to the negative ππ‘, where π΄ is equal to πΏ minus π nought over π nought. Here π represents the growth rate of the population. πΏ is the carrying capacity. And π nought is the initial population. We can identify each of these values from the information given in the question.

First, we see that π is equal to 0.07 and πΏ is equal to 900. Weβre also told that π zero is equal to 50. So we can fill in each of the values in the general solution. Letβs work out π΄ first of all. π΄ is equal to πΏ minus π nought over π nought. Thatβs 900 minus 50 over 50 or 850 over 50, which is equal to 17.

Now we can substitute into the general form of the solution. π is equal to πΏ β thatβs 900 β over one plus π΄ β thatβs 17 β π to the power of negative π β thatβs negative 0.7 β π‘. So we have our solution for π or π of π‘. Itβs equal to 900 over one plus 17π to the power of negative 0.07π‘.

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