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Unformatted text preview: McGill University
Math 325A: Diﬀerential Equations
Test 1 1. Solve the initial value problem
x + ey y (0) = 0. What is the interval of deﬁnition of your solution?
2. Find the general solution of the diﬀerential equation
= 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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