10.09.2016
Dear Student,According to our class schedule, you should complete Lectures 22, 23 and 24 and the associated homework problems by Sunday, October 23. The content summary below should help you to understand this material.
WEEK 9 -- Content Summary for Lectures 22, 23, and 24
Lecture 22 begins by introducing a special class of systems of second-order differential equations called Sturm-Liouville (or simply S-L) systems of second-order differential linear equations consisting of:
1) a second-order differential equation a[t] yâ€[t] + b[t] y’[t] +c[t] y[t] = 0 where the a[t], b[t], and c[t] are polynomials in t;
2) an interval [c, d] of real numbers.
3] an inner-product
Lectures 23 through 31 address a variety of other S-L problems including:
a) Bessel's equation of order k: t^2 y’’[t] +t y’[t] + [t^2 - k^2] y[t] = 0 where k and t are real numbers.
b) Legendre’s Equation of order n:Â Â Â Â Â Â Â Â Â [1-t^2] y”[t]-2 t y’[t] +n(n+1) y[t] where n is an integer.
A number of other such S-L systems will be applied to solve a variety of different applications in the lectures through Lecture 31, which is the last such system that will be covered on Exam 2.
It turns out that the Euler-Cauchy system is one of the easiest to solve. To see why, just let y[t] = t^k where k is a positive integer. Then, by the Chain Rule for derivatives, y’[t] = t k^(t-1), y”[t] = k(k-1) t^(k-2). Next, substitute these into the Euler-Cauchy Equation to obtain:
0=a t^2 y”[t]+b +ty’[t]+c y[t] = a t^2 (k(k-1) t^(k-2)) +b(t k t^(k-1) +c t^k = (a k^2 +(b-a) k +c ) t^k.
Therefore, y[t] = t^k is a solution of the Euler-Cauchy Equation if and only if k satisfies the auxiliary equation: a k^2 + (b-a) k + c = 0. This is the so-called direct solution method for finding all solutions of any given Euler-Cauchy Equation.
Lecture 22 goes on to discuss the indirect solution method for solving any Euler-Cauchy Equation by making the change of variables: t = e^s or, equivalently, s=Ln[t]. This change of variables transforms any Euler-Cauchy Equation into the second-order linear differential equation with constant coefficients. Both of these methods can be used to solve Euler-Cauchy Equations and both are illustrated in several examples, Just Do It!’s In Lecture 22 and homework problems for Lecture 22.
The remainder of Lecture 22 is devoted to a review of power series from your undergraduate calculus course.
Lecture 23 develops the basic power series approach and the use of power series to solve second-order linear differential equations with variable coefficients:
(*) a[t] y”[t] +b[t] y’[t] +c[t] y[t] =0
The basic facts about the Power Series Method for solving variable coefficient problems of the form (*) are:
Given any t-value t0 for which a{t0} is not zero and given any two constants, call them alpha and beta, there is one and only one solution y[t] of (*) such that y[t0] = alpha    and y’[t0] = beta. Moreover, this solution is valid on any interval containing t0 that does not contain any points where a[t0]=0. This is called the Existence-Uniqueness Theorem for Power series solutions of second-order linear differential equations with variable coefficients. Â
The following facts are consequences of the Existence-Uniqueness Theorem for power series solutions:
On any interval I of t-values for which a[t] is not 0:
1) Any second order linear differential equation (*) has two linearly independent solutions y^(1)[t] and y^(2)[t] that are valid on the interval I;
2) If y^(1)[t] and y^(2)[t] are linearly independent solutions of (*) valid on I, then any other solution y[t] of (*) can be written as a linear combination:
y[t] = c[1] y^(1)[t] + c[2] y^(2)[t] for suitable constants c[1] and c[2].
The first application of the Power Series Method to the solution of a second-order linear differential equation is to Airy’s differential equation (*): y”[t] - t y[t] = 0.
By making the substitutions: y’[t] = p a[p] t^(p-1) and y”[t] = p(p-1) a{p] t^(p-2), Airy’s Equation can be reduced to the conclusion that p=q+3.
The Power Series Method is also applied to Legendre’s Differential Equation:
((1-t^2)y”[t] -2ty’[t]} +(n(n+1))y[t] = 0
to obtain the following recurrence relation:
(p+2)(p+1) a{p+2] =((p(p-1)-n(n+1) a[p]
for the Legendre series coefficients a[p].
Using these recurrence relations and the consequences of the Existence-Uniqueness Theorem cited above, we are able to work out the basic algebraic and geometric properties of Legendre Polynomials are worked out in detail.
These properties also show how Legendre’s Differential Equations fit the general framework of an S-L system.
Lecture 24 does the corresponding power series analysis for Bessel’s Differential Equation that we have already mentioned.
The solution of Bessel’s Differential Equation requires the use of a new function called the Gamma Function. The Gamma Function is a generalization of the familiar factorial function (n-1)!. However is defined by Gamma[k]is defined for all real numbers k except 0 and the negative integers.
As we mentioned earlier, Lectures 22 through 31 deal with a variety of applications to problems in steady-state temperature distributions, electrostatic potential, heat and fluid flow among others. Each of these applications involves a new twist in the given information or application, but the mathematical approach will remain the same: Find an S-L system that models the applications and then work out the corresponding eigenvalues and eigenfunctions to solve the problem.
Watch for this recurring theme as you work your way through Lectures 22 through 31 and the associated homework problems.
Have a good week,
Tony Peressini
Cell PH: 217 840 2871
anthonyperessini@gmail.com