Singular Points

8/8/2019 Singular Points

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Regular Singular PointsMATH 365 Ordinary Differential Equations

J. Robert Buchanan

Department of Mathematics

Spring 2010

J. Robert Buchanan Regular Singular Points
http://goforward/http://find/http://goback/


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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


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.


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).


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.


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


E l


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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.


Homework


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Homework

Read Section 5.4

Exercises: 1739 odd


Singular Points

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