MATH 1005 Study Guide - Midterm Guide: C Mathematical Functions
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Math1005e test 1 - 11:35 12:25, jan. 30 2013. Consider the equation xy + y2 x2y(cid:48) = 0. (a) solve it as homogeneous equation. [4] (b) solve it as a bernoulli equation[4] x + ( y x dx u 1 = ln|x| + c u = 1 x + ( y x)2 y(cid:48) = 0 y(cid:48) = y. Solution: (a) divide by x2, so y u + u2 u(cid:48)x = u2 u(cid:48) u2 = 1 (b) divide by x2, then y(cid:48) 1. It is bernoulli equation that = 3 so u = y 2 y = u 1/2 y(cid:48) = 1. X2u 3/2u(cid:48) 2xu 1/2 = u 3/2 sin(x), multiply by u3/2, then x2u(cid:48) + 2xu = sin(x) is linear and u(cid:48) + 2 xu = sin(x) u = 1 y = u 1/2 = x2. Find the general solution of the equation y(cid:48) = 2x(y2 + 1) and its orthogonal trajectories.
