ENGR 310 - ENGINEERING ANALYSIS ILecture Notes 3 - Series SolutionsProfessor: 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 = xosingular at x = xo existence of power series solutions of a second oder ODELegendre equation, Legendre functions, Legendre polynomial,Frobenius method (extended power series method), indicial equation,hypergeometric equation, hypergeometric functionsBessel’s equation, Bessel functions of the first kind of order ν, Bessel functions of the second kind of order ν, Bessel functions of the third kind of order ν,orthogonal functions, weight, norm, orthonormalSturm-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 = 0where 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 = 0The basis of the differential equations depends on the roots of the indicial equation:- Case I: Distinct Roots do not differ by an integer:y1 = xr1 (ao + a 1 x + a2 x2 + )⋅⋅⋅y2 = xr2 (Ao + A1x + A2x2 + )⋅⋅⋅- Case II: Double Roots r1 = r2 = r = (1 - bo)/2:y1 = xr (ao + a1x2 + a2x2 + )⋅⋅⋅ y2 = xr (Ao + A1x + A2x2 + ) + y⋅⋅⋅1 lnx- Case III: Roots differ by an integar:y1 = xr1 (ao + a1x + a2x2 + )⋅⋅⋅y2 = xr2 (Ao + A1x + A2x2 + ) + k y⋅⋅⋅1 lnxIn some cases, k may be zero and the two solutions become identical to those of Case I.Proof of these basis equations are presented at the end of the chapter.■ Hypergeometric equations:x(1 - x)y” + [c - (a + b - 1)x] y’ - aby = 0* r1 = 0 and r2 = 1 - c.* y1 can be obtained from the Frobenius method and is called hypergeometric function. Itis usually designated as F(a, b, c; x), and can be written as: See also equation (18) on p. 224 for the expansion of F(a, b, c; x).* y2 is presented on p. 224, equation (19),* Many fundamental functions can be presented as hypergeometric functions, see #23, 26, 27, 28, and 29 on page 224. Page 224 also contains important properties of F, see problems 22 and 25.■ Bessel’s equation - associated with cylindrical coordinates in solving partial differential equations:x2y” + xy’ + (x2 - ν2)y = 0* indicial equation: (r + ν) (r - ν) = 0; if we choose r1 = ν ≥0 and r2 = -ν, then Jν (= y1) andJ-ν are the two elements in the basis (see series solutions, Eqs. (20) and (21) on page 229 for Jν and J-ν).* When ν is an integer, J-n = (-1)n Jn. J0 and J1 aregraphically presented in Figure 84 on p.228 and tabulated on page A97. Note Jn and J-n are linearly dependent to each other, so the second solution is yet to find.* When ν is an integer, Cases II and III in the discussion of Frobenius method have to be applied to find the second solution of the basis. This leads to the Bessel (or Neumann’s) function of the second kind of order n, Yn.1. For n = 0, Y0 and Yn are expressed in equations (6) and (8) on page 238. Note that in the derivation of Y0:Y0 = a (y2 + bJ0)where y2 = the second solution derived from Frobenius method,a = 2/πb = γ - ln 2γ = 0.577 215 664 90 = Euler constant = lim (1 + ½ + + 1/s - ln s)⋅⋅⋅ s→∞2. For n = 1, 2, 3, ..., the relation between y2,n (the second solution derived from the Frobenius method) and Yn are defined as:Yn = lim Yν ν→nwhere* The general solution for all value of ν is:y = c1 Jν + c2Yν* The first and second order Hankel’s functions of order ν (or the Bessel functions of the third kind) is defined as:Hν(1) = Jν + i YνHν (2) = Jν - iYν■ Orthogonal Sets of Functions* Definitions: orthogonality, orthonormal sets, norm, Fourier series* If function f(x) is approximated by an orthogonal sets of functions gi in the interval a tob, i.e., then the coefficients in the series can be obtained by the following integral:■ Sturm-Liouville ProblemsO.D.E. [r(x)y’]’ + [q(x) + λp(x)]y = 0B.C. k1 y(a) + k2 y(a) = 0l1 y(b) + l2 y(b) = 0* Both Legendre’s and Bessel’s equations can be reduced to a form of Sturm-Liouville equation, and, therefore, they are both special cases of the Sturm-Liouville problems. They are also considered eigenvalue problems with parameter λ.* A generalized Fourier series in which the orthogonal set is also a set of eigenfunctions, then the series is called eigenfunction expression.* Representation of olutions of a Sturm-Liouville problem by orthogonal set of