$\"\"$

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Let $\"\"$ be the population at the time $\"\"$ .

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Let $\"\"$ be the net increase per unit time.

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The rate of growth of population is given as $\"\"$ , where $\"\"$ is constant.

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$\"\"$

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In the above equation, $\"\"$ and $\"\"$ .

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The integrating factor is $\"\"$ .

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Substitute $\"\"$ .

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The general solution is $\"\"$ .

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The initial condition is $\"\"$ and $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ .

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The specific solution is $\"\"$ .

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$\"\"$ .

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$\"\"$

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The original differential equation is $\"\"$ .

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The specfic solution is $\"\"$ .

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Solve for $\"\"$ in the specfic solution.

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$\"\"$

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$\"\"$ proves the specific solution is correct.

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$\"\"$ .

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$\"\"$

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The specfic solution is $\"\"$ .