$\"\"$

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The differential equation is $\"\"$ , where $\"\"$ .

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(a)

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Find the solution of the differential equation.

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$\"\"$

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Apply Integral on each side.

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$\"\"$

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Consider $\"\"$ .

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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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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$ .

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Susbtitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Therefore, the solution of the equation is $\"\"$ .

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$\"\"$

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(b) Find the condition on $\"\"$ will lead to an exponential expansion of the population.

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The equation is $\"\"$ .

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The differential equation is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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When $\"\"$ is positive, the population $\"\"$ is increase as time $\"\"$ increases.

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Hence, $\"\"$ .

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$\"\"$

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Since $\"\"$ is positive for all positive values of $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Therefore, the population will grow when $\"\"$ .

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$\"\"$

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(c) Find the condition on $\"\"$ will result in a constant populaton and a population is decline.

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When $\"\"$ the population is constant.

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Susbtitute $\"\"$ in $\"\"$ .

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$\"\"$

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Therefore, the condition for constant population is $\"\"$ .

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When $\"\"$ the population is decline.

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Susbtitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(d) Tthe population of Ireland is $\"\"$ million .

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Hence, $\"\"$ .

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The difference between the relative birth and death rates is $\"\"$ of the population.

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The value of $\"\"$ .

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Where $\"\"$ is the birth rate and $\"\"$ is the death rate.

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$\"\"$ .

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The value of $\"\"$ is emigration constant.

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$\"\"$ .

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

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$\"\"$

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Comapre the two values $\"\"$ and $\"\"$ .

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The condition is $\"\"$ .

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Therefore, the population is declining.

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$\"\"$

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

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(b) The considtion for expansion of population is $\"\"$ .

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(c) The condition for constant population is $\"\"$ and for declining the condition is $\"\"$ .

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(d) The population is declining.