N-th order linear DE

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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
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Singular Points
Definition
Is analytic at
IF
Can be represented by power series centerd at
(i.e)
with Rgt0
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
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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
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Frobenius Theorem
Theorem 6.2
IF is a regular singular point
X2 is a regular singular point . We can
find at least one sol in the form
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Frobenius Theorem
Theorem 6.2
IF is a regular singular point
X0 is a regular singular point . We can
find at least one sol in the form
We need to find all Cn and r
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Frobenius Theorem
Theorem 6.2
IF is a regular singular point

• What is the difference between
• Frobenius Theroem
• Theorem for ordinary point

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10 points
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Frobenius Theorem
Theorem 6.2
IF is a regular singular point
Theorem 6.1
Existence of Power Series Solutions
IF is an ordinary point
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Frobenius Theorem
We need to find all Cn and r
X0 is a regular singular point . We can
find at least one sol in the form
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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
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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
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Indicial Equations ( indicial roots)
indicial equation
Find the indicial roots
indicial equation
Find the indicial roots
indicial equation
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Method of Solutions
1
Find the indicial equations and roots r1 gt r2
Case I
2
Case III
Case II
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Method of Solutions
Case III
Case II