1. Differential Equations

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

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

10. 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.

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

12. The variables x and y are connected by the differential equation

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

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

15. 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?

16. 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.

17. 2009