# Lecture Notes

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## Lecture Notes

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- Pages:
- 4
- School:
- The University of Mississippi
- Course:
- Engr 310 - Engineering Analysis I

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ENGR 310 ENGINEERING ANALYSIS I Lecture Notes 3 Series Solutions Professor Wei Yin Chen Rationale To use power series to approximate solutions of differential equations where their solutions cannot be found by the analytical methods described in the first three chapters Terminology sequence power series remainder converges at x xo for sequence converges at x xo for series diverges at x xo for sequence diverges at x xo for series analytic at x xo singular at x xo existence of power series solutions of a second oder ODE Legendre equation Legendre functions Legendre polynomial Frobenius method extended power series method indicial equation hypergeometric equation hypergeometric functions Bessel s equation Bessel functions of the first kind of order Bessel functions of the second kind of order Bessel functions of the third kind of order orthogonal functions weight norm orthonormal Sturm Liouville Problems Legendre s equation associated with spherical coordinates in solving partial differential equations General solutions equations 5 6 and 7 on p 210 Note that the solution is analytic at x 0 with radius of convergence 1 Note also that the coefficients p x and q x in the Legrendre equation are analytic at x 0 after the transformation to standard form i e y p x y q x y 0 Legrendre polynomials Coefficients of higher order terms in the general solutions of the Legendre s equation vanish when n integers thus the solutions of a Legendre s equation is called the Legendre polynomials Choices of ao and an a0 1 an 2n 2n n 2 see equations 11 and 11 and Figure 82 on p 213 Frobenius method extended power series method Rationale for the equations with singular coefficients p x or q x but they have regular singularities such that p x x and q x x2 are analytic i e y a x x y b x x2 y 0 where a x and b x are analytic at x 0 The hypergeometric equations and Bessel s equations described later are all in this form Indicial equation r2 bo 1 r co 0 The basis of the differential equations

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