Sec 6.2: Solutions About Singular Points

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Presentation transcript:

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

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

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

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

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

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

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

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

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 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

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

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

Method of Solutions Case III: Case II: