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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
+ 2y = 0,
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
x Note that the solutions y1 (x) = x−1 and y2 (x) = x−2 can be rewritten in the
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.
- Winter '14