​A separable differential equation can be solved by using the method of separation of variables. This type of differential equation can be written as ∫ g(𝑦)𝑑𝑦 = ∫ 𝑓(𝑥)𝑑𝑥.

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In real-life situations, we often encounter problems involving the use of equations with derivatives. Now, we will see how indefinite integration helps us solve these problems.

​Application in Geometry

​The indefinite integral is used to find the antiderivative function from the known derivative, therefore, using indefinite integral, we can find the equation of a curve from the equation of the slope of its tangent.

​​Example:

Given that the equation of the slope of the tangent to the curve y = f(x) at (x, y) is 𝑑𝑦/ 𝑑𝑥 = 9x² -4x. if (1, 2) is a point on the curve, find the equation of the curve.​

​Exponential Growth and Decay

​Exponential growth - describes the process of increasing an amount where the rate is proportional to the amount of a quantity present at a given instant.

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Exponential decay - describes the process of reducing an amount where the rate is proportional to the amount of a quantity present at a given instant.​

​Exponential Growth and Decay

​Some quantities grow or decay at a rate proportional to their size. This relationship can be written using the differential equation

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​ dy/dt = ky, Where k is a constant

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Which can be solved ​using the general function

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​ y(t) = y(0)e^kt​

​Example:

​Consider a colony of bacteria that is observed to grow at a rate that is proportional to the number of cells present. At the start, there are 8,000 cells. after 2 hours, it increased to 14,000. How many will there be after 1 day if the growth rate is constant

​Example:

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​After t years the population of a country is x. The rate of increase of x due to births is 0.05x, while the rate of decrease of x due to deaths is 0.03x. Write a differential equation for x. Hence, find the number of years needed for the population of the country to become twice the original population.

​Bounded and logistic Growth

​Previously, you have learned that all solutions to problems involving exponential growth (or decay) can be modeled using an equation.

This is only applicable when the growth rate is proportional to the quantity present ​at any given time.

​further, this is only true when growth is unlimited, which implies the availability of infinite natural resources. However, this is not always the case in the real world as recognized by Charles Darwin.

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When limited resources are considered, The logistic growth function​ can be used to model population growth.

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​Logistic Growth Function

​Many real-life quantities have limited growth and cannot go beyond their maximum limit L as time t increases. This relationship can be modeled using the following differential equation.

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dy/dt = ky(1 - Y/L)

The solution to this differential equation is the logistic growth function​

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y(x) = L/1 + be^-kt

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t = the time of population grows k = relative growth coefficient

y = The population after time L = carrying capacity of the ​population

​Example:

​A population of 200 deer is released into the woodland. During the first 3 years, the population increases to 500 deer. Assuming a logistic growth with a limit of 3,000 deer, find a model for the population growth and then predict the population after 8 years.

​Do you believe that population is logistic and not exponential? Justify