# Exam-Style Question on Differential Equations

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 441. This question is similar to one that appeared on an A-Level paper (specimen) for 2017. The use of a calculator is allowed.

(a) Express the following fraction in partial fractions.

$$\frac{1}{F(5-3F)}$$

The popularity of a student rock group is measured during their first year of gigs. The number of fans is modelled by the differential equation:

$$\frac{dF}{dt} = \frac{F}{15} (5-3F) \quad 0 \le t \le 12$$

where F, in hundreds, is the number of fans and t is the time measured in months since the band began performing regularly.

(b) Given that there were 100 fans when the measurements began, determine the time taken, in months, for the number of fans to increase by 50%.

(c) Show that:

$$F= \frac{A}{B+C^{-\frac{t}{3}}}$$

where A, B and C are integers to be found.

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