$\"\"$

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The differential equation is $\"\"$ .

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(a)

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Find the solution of the differential equation.

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$\"\"$

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$\"\"$

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$\"\"$

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Apply integral on each side.

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$\"\"$ Apply power rule of integral : $\"\"$ .

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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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At the initial time $\"\"$ and $\"\"$ .

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$\"\"$

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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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(b) Find the value of $\"\"$ when $\"\"$ . \ \

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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 left side of $\"\"$ then the condition is $\"\"$ .

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When there is a small positive number to a negative exponent, then the value is very positive number.

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Hence, $\"\"$ the expression approaches to $\"\"$ .

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Therefore, $\"\"$ .

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$\"\"$

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(c)

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The equation is $\"\"$ .

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The doomd day will happen at $\"\"$ .

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The number of rabits breed initially is $\"\"$ .

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The numbers of rabits breed after three months are $\"\"$ .

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The prolific breed of rabbits has the growth term is $\"\"$ .

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The function is $\"\"$ , where the growth term is $\"\"$ .

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Compare equations $\"\"$ and $\"\"$ .

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$\"\"$

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The value of $\"\"$ .Susbtitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ . $\"\"$

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$\"\"$

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

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$\"\"$

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The doomsday will happen after $\"\"$ .

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$\"\"$

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

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(b) $\"\"$ at $\"\"$ .

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(c) The droomday will happen after $\"\"$ .