Second Order Differential Equations (Edexcel A Level Further Maths: Core Pure)

Two compounds,  and , are involved in a chemical reaction. The amounts in grams of these compounds,  minutes after the reaction starts, are  and  respectively and are modelled by the differential equations

Given that  = 2 and  = 5 when  = 0

(i)

the particular solution for

(ii)

the particular solution for

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination.

The concentration of antibodies, , measured in micrograms () per millilitre () of blood, is modelled by the differential equation

where  is the number of weeks since the vaccination was given.

Initially,

• there are no antibodies in the bloodstream of the patient
• the concentration of antibodies is estimated to be increasing at 10  per week

A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than 5 .

Scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish.

At time  years, the number of white-clawed crayfish, , and the number of signal crayfish, , are modelled by the differential equations

The model predicts that, at time  years, the population of white-clawed crayfish will have died out.

Given that  = 65 and  = 85 when  = 0

find the value of , giving your answer to 3 decimal

An engineer is investigating the motion of a sprung diving board at a swimming pool.
Let  be the position of the end of the diving board when it is at rest in its equilibrium position and when there is no diver standing on the diving board.
A diver jumps from the diving board.
The vertical displacement,  , of the end of the diving board above  is modelled by the differential equation

where  seconds is the time after the diver jumps.

When  = 0, the end of the diving board is 20 cm below  and is moving upwards with a speed of 55 cm s^-1

Find, according to the model, the maximum vertical displacement of the end of the diving board above .