$\"\"$

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The differential equation is $\"\"$ ,

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where $\"\"$ is rate of growth of a population of bacteria.

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$\"\"$ is population size

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$\"\"$ is time in days $\"\"$ .

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The initial size of the population is $\"\"$ .

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After 1 day the population is $\"\"$ .

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The differential equation is $\"\"$ .

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$\"\"$

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Apply integral on each side.

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$\"\"$

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Apply Constant rule : $\"\"$ .

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$\"\"$

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Apply Power rule of integration: $\"\"$ .

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$\"\"$

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$\"\"$

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The general solution is $\"\"$ .

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$\"\"$

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The general solution is $\"\"$ .

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

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$\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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The particular solution is $\"\"$ .

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$\"\"$

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Find the value of $\"\"$ .

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

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$\"\"$

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Substitute $\"\"$ .

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

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$\"\"$

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The solution is $\"\"$ . $\"\"$

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Find the population after $\"\"$ days.

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

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$\"\"$

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$\"\"$

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The population after $\"\"$ days is $\"\"$ .