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8/8/2019 Singular Points
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Regular Singular PointsMATH 365 Ordinary Differential Equations
J. Robert Buchanan
Department of Mathematics
Spring 2010
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Singular Points
Consider the generic second order linear homogeneous ODE:
P(t)y + Q(t)y + R(t)y = 0.
If we assume that P, Q, and R are polynomials and have no
factors in common then the singular points of the ODE are the
values of t for which P(t) = 0.
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Examples
Example
Find all the singular points of the following ODEs.
1 t2(1 t2)y
+2
ty
+ 4y = 0
2 t(1 t2)3y + (1 t2)2y + 2(1 + t)y = 0
3 t2y + 2(et 1)y + (et cos t)y = 0
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Behavior of Solutions Near Singular Points
Even if t0 is a singular point, we may want to study the solution
to the ODE near t0.
Of particular importance is the behavior of a solutions y1(t) andy2(t) as t t0.
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Behavior of Solutions Near Singular Points
Even if t0 is a singular point, we may want to study the solution
to the ODE near t0.
Of particular importance is the behavior of a solutions y1(t) andy2(t) as t t0.
limtt0
|y1(t)|
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Examples
Example
1 t2y 3ty + 4y = 0; y1(t) = t2, y2(t) = t
2 ln |t|
2
t
2
y
+ 3ty
+ 5y = 0; y1(t) = t1
cos(2 ln |t|),y2(t) = t1 sin(2 ln |t|)
3 t2y + 4ty + 2y = 0; y1(t) = t1, y2(t) = t
2
4 t(t 3)y + (t + 1)y 2y = 0
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S
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Strategy
To extend the power series solution technique to singular points
we must place some restrictions on the ODE
P(t)y + Q(t)y + R(t)y = 0.
If t0 is a singular point, it may only be a weak singularity.
J. Robert Buchanan Regular Singular Points
R l Si l P i
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Regular Singular Points
DefinitionGiven the ODE
P(t)y + Q(t)y + R(t)y = 0.
for which t0 is a singular point, we say t0 is a regular singularpoint if both functions
(t t0)Q(t)
P(t)and (t t0)
2 R(t)
P(t)
are analytic (in other words have convergent Taylor series
centered at t0).
J. Robert Buchanan Regular Singular Points
R k
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Remarks
when P, Q, and R are polynomials this means the
following limits are both finite.
limtt0
(t t0)Q(t)
P(t)= p0 and lim
tt0(t t0)
2 R(t)
P(t)= q0
when P, Q, and R are polynomials this means that for tnear t0
Q(t)
P(t)
1
t t0and
R(t)
P(t)
1
(t t0)2.
any singular point which is not regular is called an
irregular singular point.
J. Robert Buchanan Regular Singular Points
E l
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Examples
Example
Determine the regular and irregular singular points for the
following ODEs.
1 t2(1 t2)y +2
ty + 4y = 0
2 t(1 t2)3y + (1 t2)2y + 2(1 + t)y = 0
3 t2y + 2(et 1)y + (et cos t)y = 0
4 t(t 3)y + (t + 1)y 2y = 0
J. Robert Buchanan Regular Singular Points
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Example
Example
Consider the following ODE for which t = 0 is a regular singularpoint.
t2
y
+ 2ty
(1 + t)y = 0
Assuming the solution is of the form y(t) =
n=0
antn find the
coefficients an.
J. Robert Buchanan Regular Singular Points
Homework
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Homework
Read Section 5.4
Exercises: 1739 odd
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