<p>A team of scientists is studying the change in population of a particular species of fish in a lake. It is observed that the rate of change of the population is proportional to the product of the population and the difference between a constant $a$ and the population. If the population of this fish at time $t$ years is $x$ thousand, it is found that the rate of change of the population is $400$ fish per year when the population is $1000$ and $600$ fish per year when the population is $3000$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{10}x\left( 5-x \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the general solution of the differential equation in part <strong>(i)</strong>, giving your answer in the form $x=\mathrm{f}\left( t \right)$.</div><p style="text-align: right;">[5]</p></li></ol>

A team of scientists is studying the change in population of a particular species of fish in a lake. It is observed that the rate of change of the population is proportional to the product of the population and the difference between a constant $a$ and the population. If the population of this fish at time $t$ years is $x$ thousand, it is found that the rate of change of the population is $400$ fish per year when the population is $1000$ and $600$ fish per year when the population is $3000$.

Show that $\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{10}x\left( 5-x \right)$.

Find the general solution of the differential equation in part **(i)**, giving your answer in the form $x=\mathrm{f}\left( t \right)$.

2022 MI J2 MYE P1 Q9

2022 MI J2 MYE P1 Q9

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