Chapter 5
Chapter 6
Power Series Solutions of Linear Differential Equations
1. Review of Properties of Power Series
2. Solutions about Ordinary Points
3. Solutions about Regular Singular Points - The Method of Frobenius
4. Bessel’s Equations and Functions
5. Legendre’s Equations and Polynomials
6. Orthogonality of Functions
7. Sturm – Liouville Theory
8. Exercises
We have seen in chapter 5 that we can solve linear differential equations of order two or more with constant coefficients. The Cauchy-Euler equation is exception. In fact most linear differential equations of higher order with variable coefficients cannot be solved in terms of elementary functions. The usual strategy for solving such type of equations is to assume a solution in the form of an infinite series and proceed in a manner similar to the method of undetermined coefficients (Section 5.6). Since these series solutions often turn out to be power series, it is appropriate to summarise properties of power series in the first section of this chapter. We conclude this chapter with the Sturm-Liouville theory dealing with eigenvalues and eigenfunctions. Strum-Liouville’s differential equation includes Bessel’s and Legendre’s equations as special cases. Examples of Strum-Liouville problems are presented.
6.1 Review of Properties of Power Series
A power series in (x-a) is an infinite series of the form
c0+ c1 (x-a) + c2 (x-a)2 +- - - - = [pic] (6.1)
Series of (6.1) is also called a power series centered at a. The power series centered at a=0 is often referred as the power series, that is, the series [pic] A power series centered at a is called convergent at a specified value of x if its sequence of partial sums SN(x) =[pic], that is, {SN (x)} is convergent. In other words the limit of {SN (x)} exists. If the limit does not exist the power series is called divergent. The set of points x at which the power series is convergent is called the interval of convergence of the power series. For R >o, a power series [pic] converges if [pic]R. If the series converges only at a then R=0, and if it converges for all x then R=(. [pic]0, sinh kl>0, so c1, = 0
This case also leads to the trivial solution, so this Sturm-Liouville problem has no negative eigenvalue.
Case (iii) ( is positive, say (=k2
Now y''+k2y=0. The auxiliary equation of this homogeneous linear differential equation with constant coefficients is
m2+k2=0. Roots are m1=ik, m2=-ik.
As discussed in Section 5.5, equation (5.18) the general solution is
y(x)=c1cos (kx) +c2 sin(kx)
Now
y(o)=c11 +c2.0=0 or c1=0
y(x)=c2 sin (kx). Finally, we need
y(l)=c2 sin kl=0
To avoid trivial solution, we need c2(0.
Then we must choose k so that sin kl=0, which means that kl must be a positive multiple of (, say kl = n(.Then
(n = [pic] for n=1,2,3,- - - -- - - -.
Each of these numbers is an eigenvalue of this Sturm-Liouville problem. Corresponding to each n, the eigenfunctions are
yn(x) = c sin [pic],
where c is any non-zero real number.
Example 6.9 Discuss solution of periodic Sturm-Liouville problem:
y''+(y=0, y(-l)=y(l), y'(-l)= y'(l)
on an interval [-l,l] for cases
i) ( = 0 (ii) ( 0
Solution: Case (i) (=0 Then y=cx+d. (See example 6.8) Now
y(-l) = - cl+d=y(l)=cl+d implies c=0. The constant function y=d satisfies both boundary conditions. Thus (=0 is an eigenvalue with nonzero constant eigenfunctions.
Case (ii) (0, say (=k2
Now as in Example 6.8 (case iii) the general solution is
y(x)=c1cos (kx)+c2 sin (kx)
Now
y(-l)=c1cos kl-c2 sin (kl)=y(l)=c1cos (kl)+c2sin (kl)
But this implies that
-c2sin (kl)=c2 sin (kl)
or –c2=c2 implying c2=0
Also y' (-l)=kc1sin (kl) + kc2 cos (kl)
= y' (l)=-kc1 sin (kl) +kc2 cos (kl).
Then
kc1 sin (kl)=0
If sin (kl)(0, then c1=c2=0, leaving the trivial solution. Thus we assume that sin kl=0 which requires that kl=n( for some positive integer n. Therefore, the numbers
(n=[pic]
are eigonvalues for n=1,2,- - - - , with corresponding eigenfunctions
yn(x)=c1 cos [pic]+c2 sin [pic]
where c1 and c2 are not both zero
6.8 Exercises:
Review of Power Series
1. Write excos x in the form of a power series. Examine whether this power series is convergent.
Solution About Ordinary Points:
Find the general solution of the following differential equations about an ordinary point in terms of two power series
2. y''-(1+x)y=0
3. y''+(cos x) y = 0
4. y''+x2y=0
5. y''+y=ex
6. y'+xy=x2-2x
7. (x2-1) y'+y=0
8. y"-(x+1) y'-y=0
Use the power series method to solve the following initial value problems
9. (x-1) y''-x y'+y=0
y(0)=-2, y' (0)=6
10. (x2+1) y''+2x y'=0, y(0)=0, y' (0)=1
11. y"+xy=0, y(0)=1, y' (0)=1
12. xy"+y+x=0, y(1)=1, y' (1)=1
Solution About Regular Singular Point: The method of Frobenius
Use the method of Frobenius to solve the following differential equations
13. x y''-x y'+y=0
14. y''+ [pic] y'-2y=0
15. x y''+ y'+y=0
16. 2x y'-3 y'-[pic]y = 0
17. 4x2y''+(3x+1)y=0
18. x y''-(x+5) y'+3y=0
19. x y''+(x-5) y'+3y=0
20. x y''+ y'+xy=0
Bessel’s Equation
Find the general solution of the following equations
21. x2 y''+x y'+(x2-1)y=0
22. x y''+x y'+xy=0
23. Verify that y=xnJn(x) is a particular solution of x y"+(1-2n) y'+xy=0, x>0
Legendre’s Equation
Solve the following equations
24. (1-x2) y''-2x y'=0
25. (1-x2) y''-2x y'+12y=0 subject to initial conditions
y(0)=0, y' (0)=1.
Sturm Liouville Theory
In each of problems 26 through 35, classify the Sturm-Liouville problem as regular, periodic, or singular; state the relevant interval, find the eigenvalues; corresponding to each eigenvalue, find an eigenfunction.
26. y''+( y=0; y' (0)= y' (l)=0
27. y''+( y=0; y(0)=0, 3y(1)+ y' (1)=0
28. y''+( y=0; y(0)=0, y' (l)=0
29. y''+( y=0; y' (0)=0, y' (l)=0
30. y''+( y=0; y' (0)=y(4)=0
31. y''+( y=0; y(0)=y((),y' (0)= y' (()
32. y''+( y=0; y(-3()=y(3(), y' (-3()= y' (3()
33. y''+( y=0; y(0)=0, y(()+2 y' (()=0
34. y'+( y=0; y(0)-2 y' (0)=0, y' (1)=0
35. y''+2 y'+(1+()y=0, y(0)=y(1)=0.
* When we replace x by [pic], the series given in (6.15) and (6.16) converge for 0 ................
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