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Unformatted text preview: Practice Problems 1 for MTH G131: Fall 2008 1). Find the general solution of the following ODE: dy dx = x 2 y (1 + x 3 ) Answer: y = p 2 / 3 ln(1 + x 3 ) + C 2). Find the solution of the following initial value problem, and determine the interval of existence: dx dt = 2 t 1 + 2 x , x (2) = 0 Answer: x =- 1 / 2 + p t 2- 15 / 4, interval of existence [ p 15 / 4 , ∞ ). 3). Newton’s law of cooling states that the surface temperature of an object changes at a rate proportional to the difference between the temperature of the object and that of its surroundings. Let u ( t ) be the object’s temperature at time t , and let u be the fixed temperature of the surroundings. Then du dt =- k | u- u | where k is a positive constant. a) Find the solution satisfying the initial condition u (0) = u 1 > u . Answer: u ( t ) = u + ( u 1- u ) e- kt . b) Suppose the temperature of a cup of tea is 200 F when freshly poured. One minute later it has cooled to 190 F in a room at 70 F. How long must elapse before the tea reaches a temperature of 150 F ?...
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- Fall '08
- Critical Point, dt, Bifurcation theory