1.08k Separable differential equations: dy/dx = f(x)g(y)

A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]

John wants to encourage more birds to come into the park near his house. Each day, starting on day 1, he puts bird food out and then observes the birds for one hour. He records the maximum number of birds that he observes at any given moment in the park each day. He believes that his observations may be modelled by the following differential equation, where \(n\) is the maximum number of birds that he observed at any given moment on day \(t\). $$\frac{dn}{dt} = 0.1n\left(1 - \frac{n}{50}\right)$$

1. Show that the general solution to the differential equation can be written in the form $$n = \frac{50A}{e^{-0.1t} + A},$$ where \(A\) is an arbitrary positive constant. [9]
2. Using his model, determine the maximum number of birds that John would expect to observe at any given moment in the long term. [1]
3. Write down one possible refinement of this model. [1]
4. Write down one way in which John's model is not appropriate. [1]

Helga invests £4000 in a savings account. After \(t\) days, her investment is worth \(£y\). The rate of increase of \(y\) is \(ky\), where \(k\) is a constant.

1. Write down a differential equation in terms of \(t\), \(y\) and \(k\). [1]
2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). [4]

It is given that \(k = \frac{r}{365}\ln\left(1 + \frac{r}{100}\right)\) where \(r\%\) is the rate of interest per annum. During the first year the rate of interest is 6% per annum.

1. Find the value of Helga's investment after 90 days. [2]

After one year (365 days), the rate of interest drops to 5% per annum.

1. Find the total time that it will take for Helga's investment to double in value. [5]

Find the general solution of the differential equation $$\frac{dy}{dx} = \frac{x}{x(1 + x^2)}$$ giving your answer in the form \(y = f(x)\). [10]

Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]