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Differential Equations

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.

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Differential Equations

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  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 °Cand 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

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