math 325

# math 325 - McGill University Math 325A: Diﬀerential...

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Unformatted text preview: McGill University Math 325A: Diﬀerential Equations Test 1 1. Solve the initial value problem dy −1 = , dx x + ey y (0) = 0. What is the interval of deﬁnition of your solution? 2. Find the general solution of the diﬀerential equation x dy = y 2 − y. dx Use your result to show that, given x0 , y0 , there is a unique solution with y (x0 ) = y0 if x0 = 0. What happens if x0 = 0? 3. A brine solution ﬂows at a constant rate of 6 L/min into a large tank which initially holds 50 liters of brine solution in which are dissolved 5 kg of salt. The solution inside the tank is kept well stirred and ﬂows out of the tank at the same rate. If the concentration of the salt in the brine entering the tank is 0.5 kg/L, determine the mass of salt in the tank at any time t. When will the concentation of salt in the tank reach 0.3 kg/L? 4. Consider the initial value problem y = x2 + y 2 , y (0) = 0. (a) Show how the fundamental existence and uniqueness theorem can be used to show that there is a unique solution y (x) on the interval |x| ≤ 1/2. (Hint: consider the rectangle |x| ≤ 1, |y | ≤ 1.) (b) Find the ﬁrst four terms a0 + a1 x + a2 x2 + a3 x3 of the Taylor series expansion of y (x) about x = 0. (c) Find the second Picard interation y2 (x). How good an approximation is it to y (x)? What can you say about yn (x), the n-th Picard interation? ...
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## This document was uploaded on 03/25/2012.

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