8.5 Cauchy-Euler Equations M3830/Date: Consider the second

M3830/Date:
8.5 Cauchy-Euler Equations
Consider the second order homogeneous Cauchy-Euler equation
(1)
ax2 y 00 + bxy 0 + cy = 0. (a, b, and c are constnats and a 6= 0.)
To find two linearly independent solutions to (1), we look for a solution of the form y = xr .
To find the constant r for which y = xr is a solution of (1), we put y = xr into (1) :
ax2
ax2 (xr )00 + bx(xr )0 + c(xr )
· r(r − 1)xr−2 + bx · rxr−1 + c · xr
[ar(r − 1) + br + c] xr
ar(r − 1) + br + c
=
=
=
=
0 for all x 6= 0
0 for all x 6= 0
0 for all x 6= 0
0
Definition: We call the equation
(2) ar(r − 1) + br + c = 0
the auxiliary, or indicial, equation for (1).
Technique: (Case 1) If (2) has two distinct real roots r1 and r2 ,
fundamental set of solutions
general solution
:
:
xr1 , xr2
y = C1 xr1 + C2 xr2
(Case 2) If (2) has two distinct imaginary roots r = α ± iβ,
fundamental set of solutions
general solution
:
:
xα cos(β ln |x|), xα sin(β ln |x|)
y = xα [C1 cos(β ln |x|) + C2 sin(β ln |x|)]
(Case 3) If (2) has one repeated root r = ro ,
fundamental set of solutions
general solution
:
:
xro , xro ln |x|
y = C1 xro + C2 xro ln |x|
For a non-homogeneous second order Cauchy-Euler equation
(4)
ax2 y 00 + bxy 0 + cy = f (x). (a, b, and c are constnats and a 6= 0.)
equivalently,
(5)
y 00 +
b 0
c
f (x)
y + 2y =
. (a, b, and c are constnats and a 6= 0.)
ax
ax
ax2
we use the variation of parameter method and write the general solution
y = u1 (x)y1 (x) + u2 (x)y2 (x)
where
• y1 (x) and y2 (x) are the two linearly independent solutions of the homogeneous case (1),
and
Z
Z
g(x)y2 (x)
g(x)y1 (x)
• u1 (x) = −
dx and u2 (x) =
dx
W (y1 , y2 )
W (y1 , y2 )
f (x)
and W = det
where g(x) =
ax2
y1 y2
y10 y20
.
EXAMPLE 1: Find a general solution of the equation 4(x + 2)2 y 00 + 5y = 0,
x > −2.
EXAMPLE 2: Find a general solution of the equation x2 y 00 + 2xy 0 − 2y = 6x−2 + 3x,
x>0

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