Integrating factor

Solve the differential equation

given that y = 3 when x = ln2

Give your answer in an exact form.

A curve passes through the point (5, 12.3) and satisfies the differential equation

Use Euler’s step by step method once, and then the midpoint formula

once, each with a step length of 0.1, to estimate the value of y when x 52

Give your answer to six significant figures.

Find the general solution of the differential equation

Given that y satisfies the differential equation in part (b)(i) and that y 123 when x 5, find the value of y when x 52

Give your answer to six significant figures.

Comment on the accuracy of your answer to part (a).

In a colony of seabirds, there are y birds at time t years.

The rate of reduction in the number of birds due to birds dying or leaving the colony is proportional to the number of birds.

In one year the reduction in the number of birds due to birds dying or leaving the colony is equal to 16% of the number of birds at the start of the year.

If no birds are born or join the colony, find the constant k such that

Give your answer to three significant figures.

A wildlife protection group takes measures to support the colony.

The rate of reduction in the number of birds due to birds dying or leaving the colony is the same as in part (a), but in addition:

The rate of increase in the number of birds due to births is 20t per year.

The wildlife protection group brings 45 birds into the colony each year.

Write down a first-order differential equation for y and t

The initial number of birds is 340

Solve your differential equation from part (b) to find y in terms of t

Describe two limitations of the model you have used.

Find the general solution of the differential equation

Find the particular solution for which y 0 when x 3

Give your answer in the form y f (x)

Find the general solution of the differential equation