Cauchy-Euler Differential Equation
A linear differential equation of the form
n 1
dny
y
dy
n 1 d
an x

a
x


a
x
 a0 yg g( x()x)

n 1
1
dx n
dx n1
dx
n
Homogeneous g ( x)  0
dny
d n 1 y
dy
an n  an 1 n 1    a1  a0 y0 0
dx
dx
dx
We assumed a solution of the form y  e mx .
n 1
dny
y
dy
n 1 d

an x

a
x


a
x
 a0 y00
n 1
1
dx n
dx n1
dx
n
We assume a solution of the form y  x .
mx
G. Carbone MATH-282HM
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Cauchy-Euler Differential Equation
Review – Of a Second Order DE with
Constant Coefficients.
ay  by  cy  0
am 2  bm  c  0
y  c1e m1x  c2 e m2 x
y  c1e m1x  c2 xe m1x
y  e  c1 cos(  x)  c2 sin(  x) 
Where: b 
2a
b 2  4ac
   i
2a
G. Carbone MATH-282HM
b 2  4ac  0
2
Cauchy-Euler Differential Equation
a2 x 2 y   a1 xy   a0 y  0
y  x m , y  mx m1 , y   m(m  1) x m 2
a2 x 2 m(m  1) x m2  a1 xmx m1  a0 x m  0
a2 m(m  1) x m  a1mx m  a0 x m  0
 a ( m  m)  a m  a  x  0
2
2
m
1
0
a2 (m  m)  a1m  a0  0
2
x 2 y  5 xy  3 y  0
Case I, Example
a2  1, a1  5, a0  3
Roots are real and
distinct
a  1, b  4, c  3
m 2  4m  3  0
a2 m  (a1  a2 )m  a0  0
(m  1)(m  3)  0
a2  a, b  a1  a2, c  a0
m1  1, m2  3
2
or
4
16  4 1  3

2
2
m  2  1, m1  1, m2  3
m
y  c1 x 1  c2 x 3
Check
G. Carbone MATH-282HM
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Cauchy-Euler Differential Equation
Example
Case II Roots are equal.
x 2 y  5 xy   4 y  0
y  5x y  42 y  0
x
a  1, b  4, c  4
m  4m  4  0
2
y2  x
2
y2  x
2
(m  2)(m  2)  0
m1  2, m2  2
y  c1 x
2
We need a second
solution
y1 ( x)  x 2 , P ( x)  5x
 5x dx

5ln( x )
e 
2 e
dx  x
dx
( x 2 ) 2
x 4

x 5
2 1
dx

x
dx  x 2 ln( x )
4
x
x


y2  c2 x 2 ln( x)
y  c1 x 2  c2 x 2 ln( x ) x  0
Check
G. Carbone MATH-282HM
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Cauchy-Euler Differential Equation
Case II Roots are equal.
In general
a
x
a
y  a 1x y  02 y  0
2
b  a1  a2 , 0= b 2  4ac
y1  x m ,
m  b  a22 aa1
2a2
2
y2  x m
y12  ( x m ) 2  x 2 m  x ( a2 a1 )/ a2
a
P ( x)  a21x
e
 aa12 ln( x )
e
ln( x  a1/ a2 )
 xa a
1/ 2
y2  x
m
y2  x m
G. Carbone MATH-282HM
a
 a 1x dx
2
e 
 (x )
m 2
dx  x m
xa a
1/ 2

 aa12 ln( x )
e
dx
m 2
(x )
x a a
 x(a a )/ a dx  x  x  x
2

1
m
2
1/ 2
 a1/ a2
dx
1
dx  x m ln( x)
x
5
Cauchy-Euler Differential Equation
y  k1 x i  k2 x i  k1 x x i  k2 x x  i
a2 x 2 y   a1 xy   a0 y  0
y  x m , y  mx m1 , y   m(m  1) x m 2
a2 x m(m  1) x
2
m2
 a1 xmx
m 1
 a0 x  0
a2 m(m  1) x m  a1mx m  a0 x m  0
 a ( m  m)  a m  a  x  0
2
2
m
1
0
a2 (m 2  m)  a1m  a0  0
a2 m 2  (a1  a2 )m  a0  0
a2  a, b  a1  a2, c  a0
m
x i  ei ln x ,
We make use of
Euler’s Equation
x  i  e  i ln x
ei  cos( )  i sin( )
   ln x
y  k1 x  cos(  ln x )  i sin(  ln x)   k 2 x  cos(  ln x)  i sin(  ln x) 
y  x  k1 cos(  ln x)  k3 sin(  ln x)   x  k2 cos(  ln x )  k4 sin(  ln x) 
y  x  (k1  k2 ) cos(  ln x)  (k3  k4 )sin(  ln x) 
y  x  c1 cos(  ln x )  c2 sin(  ln x ) 
G. Carbone MATH-282HM
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Cauchy-Euler Differential Equation
Example:
a2 x 2 y  a1 xy  a0 y  0

y  x 1 c1 cos( 26 ln x )  c2 sin( 26 ln x)

2 x 2 y   6 xy   5 y  0
a2  a  2
b  a1  a 2  6  2  4
a0  c  5
b
b 2  4ac
m

2a
2a
m
4
16  40
24

 1 
22
22
4
 1 
2 6i
6i
 1 
4
2
G. Carbone MATH-282HM
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