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QLDSpecialist MathematicsQuick questions

Unit 4: Further calculus, and statistical inference

Quick questions on First-order differential equations and separation of variables (QCE Specialist Mathematics Unit 4)

6short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are separation of variables?
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If a first-order equation can be written as dydx=f(x)g(y)\dfrac{dy}{dx} = f(x)g(y), separate the variables so each side involves only one variable, then integrate both sides:
What is newton's law of cooling?
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The rate of cooling is proportional to the temperature difference from the surroundings TsT_s:
What is the logistic model?
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When growth is limited by a carrying capacity MM, the model is
What is solve for PP?
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Exponentiating, P=Ae0.2tP = A e^{0.2t} where A=eCA = e^{C}.
What is apply the initial condition?
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At t=0t = 0, P=500P = 500, so A=500A = 500. Thus
What is check?
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Substituting t≈6.93t \approx 6.93: 500 e0.2×6.93=500 e1.386=500×4.00=2000500\,e^{0.2 \times 6.93} = 500\,e^{1.386} = 500 \times 4.00 = 2000, as required.

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