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Solutions of N-th Order Linear Differential Equations via Power Series

This section focuses on the solutions of N-th order linear differential equations using power series, particularly centered around ordinary points. It covers various methods, including the Cauchy-Euler equation, the non-homogeneous approach, and the annihilator method. We will find two linearly independent power series solutions about ( x=0 ), exploring definitions of ordinary and singular points. Theorems related to the existence of power series solutions at ordinary points will also be examined, emphasizing cases with polynomial coefficients.

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Solutions of N-th Order Linear Differential Equations via Power Series

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  1. CH 6: Series Solutions of Linear Equations N-th order linear DE Constant Coeff variable Coeff Cauchy-Euler 4.7 In General Ch 6 Power Series Homog(findyp) 4.3 NON-HOMOG (find yp) Annihilator Approach 4.5 Variational of Parameters 4.6

  2. WED

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  4. Sec 6.2: Solution about Ordinary Points Find two power series solutions of the given differential equation about the point x=0

  5. Ordinary 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

  6. Ordinary Points 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

  7. Ordinary Points Special Case: Polynomial Coefficients

  8. Existence of Power series Solutions Theorem 6.1: Existence of Power Series Solutions IF is an ordinary point ? X=0 is an ordinary point . We can find 2-lin-indep sol in the form

  9. Existence of Power series Solutions Theorem 6.1: Special case IF is an ordinary point

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