# Fortunately there is a trick that enables us to

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Unformatted text preview: is quite simple in the case of Cauchy-Euler equations. Indeed, in this case, we can simply take y (x) = xr , substitute into the equation and solve for r. Often there will be two linearly independent solutions y1 (x) = xr1 and y2 (x) = xr2 of this special form. In this case, the general solution is given by the superposition principle as y = c1 xr1 + c2 xr2 . For example, to solve the diﬀerential equation x2 d2 y dy + 2y = 0, + 4x 2 dx dx we set y = xr and diﬀerentiate to show that dy/dx = rxr−1 d2 y/dx2 = r(r − 1)xr−2 ⇒ x(dy/dx) = rxr , ⇒ x2 (d2 y/dx2 ) = r(r − 1)xr . Substitution into the diﬀerential equation yields r(r − 1)xr + 4rxr + 2xr = 0, and dividing by xr yields r(r − 1) + 4r + 2 = 0 or r2 + 3r + 2 = 0. The roots to this equation are r = −1 and r = −2, so the general solution to the diﬀerential equation is y = c1 x−1 + c2 x−2 = c1 c2 + 2. x x Note that the solutions y1 (x) = x−1 and y2 (x) = x−2 can be rewritten in the form y1 (x) = x−1 ∞ y2 (x) = x−2 an xn , n=0 ∞ bn xn , n=0 where a0 = b0 = 1...
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## This document was uploaded on 01/12/2014.

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