$\"\"$

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The differential equation is $\"\"$ , where $\"\"$ is a constant and $\"\"$ is the carrying capacity.

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(a) Solve the differential equation.

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$\"\"$

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$\"\"$

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Apply integral on each side.

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$\"\"$

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$\"\"$

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Let $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the integral.

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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The solution of differential equation is $\"\"$ .

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$\"\"$

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(b) Find $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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As $\"\"$ tends to $\"\"$ , $\"\"$ approaches to $\"\"$ .

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$\"\"$ .

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$\"\"$

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(c) Graph the function $\"\"$ when $\"\"$ and $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ .

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From example (2) the logistic differential equation is $\"\"$ .

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Graph the functions $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph:

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The two functions have the same initial populations and same equilibrium positions.

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The two functions have different points of inflection.

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$\"\"$

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(d)

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The logistic differential equation : $\"\"$ .

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$\"\"$

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Implicitly differentiate with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The population will be growing fastest when $\"\"$ reaches to maximum.

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Find the maximum value equate $\"\"$ to zero.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(a) The solution of differential equation is $\"\"$ .

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(b) $\"\"$ .

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(c) Graph of the functions $\"\"$ and $\"\"$ .

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$\"\"$

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The two functions have the same initial populations and same equilibrium positions.

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The two functions have different points of inflection.

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(d) $\"\"$ .