Differential Equations

Write the differential equation

• P is the pressure in a gas-filled balloon and V is the volume of the balloon.

• The rate at which P changes, as V changes, is inversely proportional to the square of the volume of the balloon.

• P is the pressure in a gas-filled balloon and V is the volume of the balloon.

• The rate at which P changes, as V changes, is inversely proportional to the square of the volume of the balloon.

A boat is being tested in various sea conditions to see what happens when the engine is turned off.

• In rough conditions, the boat slows at a rate proportional to the square of its velocity v and inversely proportional to its mass m. Write down a differential equation (for the velocity) which models the boat's motion.

A boat is being tested in various sea conditions to see what happens when the engine is turned off.

• In smooth conditions, a differential equation for the boat's velocity is

• Find the general solution of this equation.

A boat is being tested in various sea conditions to see what happens when the engine is turned off.

• In smooth conditions, a differential equation for the boat's velocity is

• Find the general solution of this equation.

• If the boat has an initial velocity of 5 ms-1 show that it cannot travel more than 500 m from its initial position.

1991

• A model for the way in which news, being broadcast regularly by a radio station, is spread throughout a region with population P is given by

• Here t is time, k is a positive constant and n is the number of people who have heard the news.

• Solve the differential equation to obtain ,

• given that when t = 0, n = 0 too. Show all steps in solving the equation; no marks will be given for just showing that this solution satisfies the equation.

• Given that 50% of the population have heard the news after 5 hours, find by which time 90% of the population have heard.

A heavy wooden beam 6 m long, with a rectangular cross section, is supported at each end only, so it bends to take on a

slightly curved shape.

A point W on the beam is x metres horizontally from A and y metres vertically below the line AB.

The variables x and y are connected by the differential equation

Solve this differential equation to find a formula for the sag y in the beam in terms of x.

Information for evaluating the constants: x = 0, y = 0; x = 6, y = 0

2011

• In radioactive decay, a radioactive substance decays at a rate proportional to the number of radioactive atoms present. This can be modelled by the differential equation

• dN/dt = k N • where N is the number of radioactive atoms present and t is

time in days. • Iodine 131 is a radioactive isotope of iodine. • Iodine 131 has a half life of 8.0 days (ie after 8 days half of any

atoms of iodine 131 present would have decayed). • A nuclear accident produces a quantity of iodine 131.

How long after the accident will it take for 99% of the iodine 131 to decay?

2010• James is baking a cake.

When he takes the cake out of the oven, the temperature of the cake is 180°C. James puts it on a cake rack in the kitchen.After one hour the cake has cooled to 100°C.It needs to cool to 35°C before it can be iced.The rate of cooling of the cake can be modelled by the differential equation

• dT/dt = k(T − 20) • where T is the temperature of the cake in °C

and t is the time in hours after the cake was taken out of the oven.

• Solve the differential equation to find the minimum time James needs to leave the cake before he can ice it.

2009