Question

Let the population of rabbits surviving at a time t be governed by the differential equation. $fraction numerator dp left parenthesis straight t right parenthesis over denominator dt end fraction space equals space 1 half straight p left parenthesis straight t right parenthesis space minus 200$ If p(0) = 100 then p(t) is equal to

$400 minus 300 straight e to the power of straight t over 2 end exponent$
$300 minus 200 straight e to the power of negative straight t over 2 end exponent$
$600 minus 500 straight e to the power of straight t over 2 end exponent$
$400 minus 300 straight e to the power of negative straight t over 2 end exponent$

Solution

400 minus 300 straight e to the power of straight t over 2 end exponent

Given differential equation $dp over dt minus 1 half straight p left parenthesis straight t right parenthesis space equals space minus 200$ is a linear differential equation
Here, p(t) = $fraction numerator negative 1 over denominator 2 end fraction comma space straight Q space left parenthesis straight t right parenthesis space equals space minus space 200$
$If space equals space straight e to the power of integral negative open parentheses 1 half close parentheses dt end exponent space equals space straight e to the power of negative 1 half end exponent$
Hence, solution is
p(t), IF = ∫Q(t)IF dt
$straight p left parenthesis straight t right parenthesis. straight e to the power of negative straight t over 2 end exponent space equals space integral negative 200. straight e to the power of negative straight t over 2 end exponent dt straight p left parenthesis straight t right parenthesis. straight e to the power of negative straight t over 2 end exponent space equals integral negative 200 space straight e to the power of negative straight t over 2 end exponent. dt straight p left parenthesis straight t right parenthesis. straight e to the power of negative straight t over 2 end exponent space equals space 400 space straight e to the power of negative straight t over 2 end exponent space plus space straight K rightwards double arrow space straight p left parenthesis straight t right parenthesis space equals space 400 space plus ke to the power of negative 1 half end exponent If space straight p left parenthesis 0 right parenthesis space equals space 100 space comma then space straight k space equals negative 300 rightwards double arrow space straight p left parenthesis straight t 0 space equals space 400 space minus 300 to the power of straight t over 2 end exponent$

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