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MATH 557 Homework Set #1 Fall 2015 1. Give a correct explanation of why the separation of variables method works for solving dx P (t) = dt Q(x) 2. If we use the separation of variables method to solve dx = x2 dt do we obtain all possible solutions? Discuss. 3. Find the general solution of the third order Euler type equation (t − 1)3 φ000 + 8(t − 1)2 φ00 + 14(t − 1)φ0 + 6φ = 0 4. If a, b, c > 0 show that any solution of aφ00 + bφ0 + cφ = 0 satisfies limt→+∞ φ(t) = 0. 5. An ODE is said to be of Bernoulli type if it can be written as φ0 + a(t)φ = b(t)φn for some real exponent n. Show that the Bernoulli equation can be transformed to a first order linear equation by means of the change of variable ψ = φ1−n . Find the general solution of tφ0 + φ = t4 φ3 1 1 . (It is not too 6. Find the general solution of the system φ = Aφ where A = 0 1 hard to compute etA directly from its power series definition in this case.) 0