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ay′′ + by′ + cy = 0
2nd Order Homogeneous Linear Equation with Constant Coefficients
set equal
to zero
y′′ is the
highest
derivative
y and all
a, b, and c
derivatives
are constant
are raised to
the first power
( no ( y′) )
2
Assume y = erx , r constant
cy = ce rx
by′ = bre
rx
never
zero
ay′′ = ar 2 e rx
0 = ar 2 e rx + bre rx + ce rx ⇒ e
rx
(
Auxiliary (or Indicial)
Equation
The solution is
based on the
ar 2 + br + c = 0
roots of this
equation
)
1
ay′′ + by′ + cy = 0
2
⇓
roots r1 and r2
ar + br + c
based on b 2 − 4ac
i real distinct roots r1 and r2 ( b 2 − 4ac > 0 )
y = c1e r1x + c2 er2 x
i repeated real roots r = r ( b
y = c1e r1x + c2 xer1x
1
2
2
− 4ac = 0 )
i complex roots r1 = α + β i, r2 = α − β i ( b 2 − 4ac < 0 )
y = eα x ( c1 cos ( β x ) + c2 sin ( β x ) )
ax 2 y′′ + bxy′ + cy = 0
Cauchy-Euler Equation
2nd Order Homogeneous Linear Equation with Constant Coefficients
Variable
with the degree on x = order of the derivative
Assume y = x r , r constant
cy = cx r
⇒ bxy′ = brx r
⇒ ax 2 y′′ = ar ( r − 1) x r
bxy′ = bx ( rx r −1 )
ax 2 y′′ = ax 2 r ( r − 1) x r − 2
⇒ x r ( ar 2 + ( b − a ) r + c ) = 0
In order for a solution to exist,
we need ax 2 ≠ 0 ( see Thm. 3.1)
⇒ x ≠ 0 and our solution
xr ≠ 0
will be valid on the interval ( 0, ∞ )
on ( 0, ∞ )
Auxiliary Equation
The solution is
based on the
roots of this
equation
2
ax 2 y′′ + bxy′ + cy = 0
⇓
roots r1 and r2
ar 2 + ( b − a ) r + c
2
based on ( b − a ) − 4ac
i real distinct roots r1 and r2
r1
r2
1
2
y =c x +c x
i repeated real roots r1 = r2
y = c1 x r1 + c2 x r1 ln x
i complex roots r1 = α + β i, r2 = α − β i
y = xα ( c1 cos ( β ln x ) + c2 sin ( β ln x ) )
Why?
The substitution x = et reduces our variable coefficient equation
ax 2 y′′ + bxy′ + cy = 0 into a constant coefficient equation in t.
To see this, lets use
dy
d2y
and
in place of y ′ and y′′.
dx
dx 2
d2y
dy
ax
+
bx
+ cy = 0
2
dx
dx
2
x = et
dy dy dx
dy dy t
dy
dy
=
⇒
= e ⇒
= e−t
dt dx dt
dt dx
dx
dt
chain rule
3
d 2 y d  dy 
=  
dt 2 dt  dt 
=
d  dy dx 
 ⋅ 
dt  dx dt 
Use the product rule
 d  dy  dx   dx   dy   d 2 x 
=    ⋅   +    2 
 dx  dx  dt   dt   dx   dt 
( derivative of first w.r.t t ) ( second )
( first )
d 2 y   d 2 y  t  t  dy  t
= 
 ⋅ e  (e ) +   (e )
dt 2   dx 2  
 dx 
derivative of
the second
w.r.t. t
d 2 y   d 2 y  t  t  − t dy  t
= 
 ⋅ e  (e ) +  e
 (e )
dt 2   dx 2  
dt 

d 2 y dy  d 2 y  2t
−
=
⋅e
dt 2 dt  dx 2 
2
d2y
dy 
−2 t  d y
=
e
−


2
dx 2
dt 
 dt
2
ax 2
d y
dy
+
bx
+ cy = 0
2
dx
dx
⇓
dy
dy
= e−t
dx
dt
2
2
d y
dy 
−2 t  d y
=
e
 2 − 
2
dx
dt 
 dt

 d 2 y dy  
 dy 
ae 2t  e −2t  2 −   + bet  e −t
 + cy = 0
dt  
dt 

 dt

⇓
2
a
d y
dy
+ ( b − a ) + cy = 0
2
dt
dt
constant coefficients
4
x = et
⇒ t = ln x
i complex roots r1 = α + β i, r2 = α − β i
i real distinct roots r1 and r2
y = eαt (c1 cos(β t ) + c2 sin (β t ))
y = c1e r1t + c2e r2t
y = c1e
r1 ln x
+ c2 e
r2 ln x
r1
y = c1e ln x + c2 e ln x
y = eα ln x (c1 cos(β ln x ) + c2 sin (β ln x ))
r2
α
y = e ln x (c1 cos(β ln x ) + c2 sin (β ln x ))
y = c1 x r1 + c2 x r2
y = xα ( c1 cos ( β ln x ) + c2 sin ( β ln x ) )
i repeated real roots r1 = r2
y = c1e r1t + c2te r1t
y = c1e r1 ln x + c2 ln xe r1 ln x
r1
y = c1e ln x + c2 ln xe ln x
r1
y = c1 x r1 + c2 x r1 ln x
3.6 # 8
x 2 y ′′ + 3 xy ′ − 4 y = 0 → Cauchy-Euler
y = x m ⇒ y′ = mx m −1 ⇒ y ′′ = m ( m − 1) x m − 2
x 2 y ′′ = m ( m − 1) x m − 2 x 2 = m ( m − 1) x m
3 xy′ = 3mx m −1 x = 3mx m
−4 y = −4 x m
0 =  m ( m − 1) + 3m − 4  x m ⇒ m 2 + 2m − 4 = 0
0
m=
−2 ± 4 − 4 (1)( −4 )
2
=
−2 ± 20 −2 ± 2 5
=
= −1 ± 5
2
2
Real Distinct Roots ⇒ y = c1 x −1+
5
+ c2 x −1−
5
5
3.6 # 16
x 3 y′′′ + xy′ − y = 0 → Cauchy-Euler
y = x m ⇒ y′ = mx m−1 ⇒ y′′ = m ( m − 1) x m −2 ⇒ y′′′ = m ( m − 1)( m − 2 ) x m−3
x 3 y′′′ = m ( m − 1)( m − 2 ) x m−3 x3 = m ( m − 1)( m − 2 ) x m
xy ′ = mx m−1 x = mx m
− y = − xm
3
0 =  m ( m − 1)( m − 2 ) + m − 1 x m ⇒ m3 − 3m 2 + 3m − 1 = 0 ⇒ ( m − 1) = 0
0
m = 1, Repeated Root with multiplicity 3
⇒ y = c1 x + c2 x ln x + c3 x ( ln x )
2
6