1. Sec 6.2: Solutions About Singular Points
N-th order linear DE
Constant Coeff
variable Coeff
Cauchy-Euler
4.7
Ch 6 Series
Point
Homog(find yp)
4.3
NON-HOMOG
(find yp)
Annihilator
Approach
4.5
Variational of
Parameters
4.6
Ordinary
6.1
Singular
6.2

2. Singular Points Definition: Is analytic at IF:
Can be represented by power series centerd at
(i.e)
with R>0
Definition:
Is an ordinary point of the DE (*)
IF:
are analytic at
A point that is not an ordinary point of the DE(*) is said to be singular point
Special Case:
Polynomial Coefficients

3. Regular Singular Points
Definition:
Is a regular singular point of the DE (*)
IF:
are analytic at
A singular point that is not a regular singular point of the DE(*) is said to be irregular singular point

4. Frobenius’ Theorem Theorem 6.2: IF is a regular singular point
X=2 is a regular singular point .
We can find at least one sol in the form

5. We need to find all Cn and r
Frobenius’ Theorem
Theorem 6.2:
IF is a regular
singular point
X=0 is a regular singular point .
We can find at least one sol in the form
We need to find all Cn and r

6. 3 Frobenius’ Theorem 10 points What is the difference between
IF is a regular
singular point
What is the difference between
Frobenius Theroem
Theorem for ordinary point 3 10
points

7. Frobenius’ Theorem Theorem 6.2: IF is a regular singular point
Existence of Power Series Solutions
IF is an
ordinary point

8. We need to find all Cn and r
Frobenius’ Theorem
We need to find all Cn and r
X=0 is a regular singular point .
We can find at least one sol in the form

9. Indicial Equations ( indicial roots)
indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero
indicial equation
indicial roots

10. Indicial Equations ( indicial roots)
indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero
indicial equation
indicial roots
indicial equation

11. Indicial Equations ( indicial roots)
Find the indicial roots:
indicial equation
Find the indicial roots:
indicial equation

12. Method of Solutions 1
Find the indicial equations and roots: r1 > r2
Case I: 2
Case III:
Case II:

13. Method of Solutions
Case III:
Case II: