5.4 Bessel s Equation. Bessel Functions

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1 SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent variable Find a general solution in terms of hypergeometric functions x t t 5 x( x)ys ( 6x)yr y t t 6 x( x)ys ( x)yr y and parameters related by Ct D c(t t ), 7 4x( x)ys C a b, K ab From this you see that (5) 8 4(t 3t )y ## yr y # 8y y is a normalized form of the more general (8) and that various cases of (8) can thus be solved in terms 9 (t ## # 5t 6)y (t 3)y 8y of hypergeometric functions 3t( t)y ## ty # y 54 Bessel s Equation Bessel Functions J (x) One of the most important ODEs in applied mathematics in Bessel s equation, 6 () x ys xyr (x )y where the parameter (nu) is a given real number which is positive or zero Bessel s equation often appears if a problem shows cylindrical symmetry, for example, as the membranes in Sec9 The equation satisfies the assumptions of Theorem To see this, divide () by x to get the standard form ys yr>x ( >x )y Hence, according to the Frobenius theory, it has a solution of the form () y(x) a m a m x mr (a ) Substituting () and its first and second derivatives into Bessel s equation, we obtain a m (m r)(m r )a m x mr a m (m r)a m x mr a We equate the sum of the coefficients of to zero Note that this power corresponds to m s in the first, second, and fourth series, and to m s in the third series Hence for s and s, the third series does not contribute since m m x sr a m x mr a a m x mr m x sr 6 FRIEDRICH WILHELM BESSEL ( ), German astronomer and mathematician, studied astronomy on his own in his spare time as an apprentice of a trade company and finally became director of the new Königsberg Observatory Formulas on Bessel functions are contained in Ref [GenRef] and the standard treatise [A3]

2 88 CHAP 5 Series Solutions of ODEs Special Functions For s, 3, Á all four series contribute, so that we get a general formula for all these s We find (a) r(r )a ra a (s ) (3) (b) (c) (r )ra (r )a a (s r)(s r )a s (s r)a s a s a s (s ) (s, 3, Á ) From (3a) we obtain the indicial equation by dropping, (4) (r )(r ) The roots are r ( ) and r Coefficient Recursion for r r v For r, Eq (3b) reduces to ( )a Hence a since Substituting r in (3c) and combining the three terms containing gives simply a s a (5) (s )sa s a s Since a and, it follows from (5) that a 3, a 5, Á Hence we have to deal only with even-numbered coefficients with s m For s m, Eq (5) becomes Solving for a m (m )ma m a m gives the recursion formula a s (6) a, m,, Á m m( m) a m From (6) we can now determine a, a 4, Á successively This gives and so on, and in general a 4 a () a a (v ) a 4! ( )( ) (7) a m () m a m m! ( )( ) Á (m), m,, Á Bessel Functions J n (x) for Integer n Integer values of v are denoted by n This is standard For nthe relation (7) becomes () m a (8) a m,, Á m m m! (n )(n ) Á (n m),

3 SEC 54 Bessel s Equation Bessel Functions J (x) 89 a is still arbitrary, so that the series () with these coefficients would contain this arbitrary factor a This would be a highly impractical situation for developing formulas or computing values of this new function Accordingly, we have to make a choice The choice a would be possible A simpler series () could be obtained if we could absorb the growing product (n )(n ) Á (n m) into a factorial function (n m)! What should be our choice? Our choice should be (9) a n n! because then n! (n ) Á (n m) (n m)! in (8), so that (8) simply becomes () m () a m,, Á m mn m! (n m)!, By inserting these coefficients into () and remembering that c, c 3, Á a particular solution of Bessel s equation that is denoted by J n (x): we obtain () J n (x) x n a m () m x m mn m! (n m)! (n ) J n (x) is called the Bessel function of the first kind of order n The series () converges for all x, as the ratio test shows Hence J n (x) is defined for all x The series converges very rapidly because of the factorials in the denominator EXAMPLE Bessel Functions J (x) and J (x) For n we obtain from () the Bessel function of order () () m x m J (x) a x x 4 x 6 Á m (m!) (!) 4 (!) 6 (3!) m which looks similar to a cosine (Fig ) For n we obtain the Bessel function of order () m x m (3) J (x) a, m m! (m )! x x 3 3!! x 5 5! 3! x 7 7 3! 4! Á m which looks similar to a sine (Fig ) But the zeros of these functions are not completely regularly spaced (see also Table A in App 5) and the height of the waves decreases with increasing x Heuristically, n >x in () in standard form [() divided by x ] is zero (if n ) or small in absolute value for large x, and so is yr>x, so that then Bessel s equation comes close to ys y, the equation of cos x and sin x; also yr>x acts as a damping term, in part responsible for the decrease in height One can show that for large x, (4) J n (x) B px cos ax np p 4 b where is read asymptotically equal and means that for fixed n the quotient of the two sides approaches as x :

4 9 CHAP 5 Series Solutions of ODEs Special Functions J 5 J 5 x Fig Bessel functions of the first kind J and J Formula (4) is surprisingly accurate even for smaller x () For instance, it will give you good starting values in a computer program for the basic task of computing zeros For example, for the first three zeros of J you obtain the values 356 (45 exact to 3 decimals, error 49), 5498 (55, error ), 8639 (8654, error 5), etc Bessel Functions J (x) for any Gamma Function We now proceed from integer n to any We had a >( n n!) in (9) So we have to extend the factorial function n! to any For this we choose (5) a ( ) with the gamma function ( ) defined by (6) ( ) e t t dt ( ) (CAUTION! Note the convention on the left but in the integral) Integration by parts gives ( ) e t t ` e t t dt () This is the basic functional relation of the gamma function (7) ( ) () Now from (6) with and then by (7) we obtain () e t dt e t ` () and then () # ()!, (3) ()! and in general (8) (n ) n! (n,, Á )

5 SEC 54 Bessel s Equation Bessel Functions J (x) 9 Hence the gamma function generalizes the factorial function to arbitrary positive Thus (5) with n agrees with (9) Furthermore, from (7) with given by (5) we first have () m a m m m! ( )( ) Á (m) ( ) Now (7) gives ( )( ) ( ), ( )( ) ( 3) and so on, so that ( )( ) Á (m)( ) ( m ) Hence because of our (standard!) choice (5) of (9) a m () m m m! ( m ) the coefficients (7) are simply With these coefficients and r r we get from () a particular solution of (), denoted by J (x) and given by () J (x) x () m x m a m m m! ( m ) J (x) is called the Bessel function of the first kind of order The series () converges for all x, as one can verify by the ratio test Discovery of Properties from Series a Bessel functions are a model case for showing how to discover properties and relations of functions from series by which they are defined Bessel functions satisfy an incredibly large number of relationships look at Ref [A3] in App ; also, find out what your CAS knows In Theorem 3 we shall discuss four formulas that are backbones in applications and theory a THEOREM Derivatives, Recursions The derivative of J (x) with respect to x can be expressed by J (x) or J (x) by the formulas () (a) [x J (x)]r x J (x) (b) [x J (x)]r x J (x) Furthermore, J (x) and its derivative satisfy the recurrence relations () (c) (d) J (x) J (x) x J (x) J (x) J (x) Jr (x)

6 9 CHAP 5 Series Solutions of ODEs Special Functions PROOF (a) We multiply () by and take under the summation sign Then we have We now differentiate this, cancel a factor, pull out, and use the functional relationship ( m ) ( m)( m) [see (7)] Then () with instead of shows that we obtain the right side of (a) Indeed, (x J )r a m (b) Similarly, we multiply () by, so that in () cancels Then we differentiate, cancel m, and use m! m(m )! This gives, with m s, (x J )r a m x x J (x) a m x () m (m )x m m m! ( m ) x () m x m () m x m m m! ( m ) Equation () with instead of and s instead of m shows that the expression on the right is x J (x) This proves (b) (c), (d) We perform the differentiation in (a) Then we do the same in (b) and multiply the result on both sides by x This gives (a*) x J x Jr x J (b*) x J x Jr x J Substracting (b*) from (a*) and dividing the result by (b*) and dividing the result by x gives (d) x x x x a m m (m )! ( m ) a s x () m x m m m! ( m) () s x s s s! ( s ) gives (c) Adding (a*) and EXAMPLE Application of Theorem in Evaluation and Integration Formula (c) can be used recursively in the form for calculating Bessel functions of higher order from those of lower order For instance, J (x) J (x)>x J (x), so that J can be obtained from tables of J and J (in App 5 or, more accurately, in Ref [GenRef] in App ) To illustrate how Theorem helps in integration, we use (b) with 3 integrated on both sides This evaluates, for instance, the integral A table of J 3 I J (x) x J (x) J (x) x 3 J 4 (x) dx x 3 J 3 (x) (on p 398 of Ref [GenRef]) or your CAS will give you 8 J 3() J 3 () # Your CAS (or a human computer in precomputer times) obtains J 3 from (), first using (c) with, that is, J then (c) with, that is, J x 3 4x J J, J J Together,

7 SEC 54 Bessel s Equation Bessel Functions J (x) 93 This is what you get, for instance, with Maple if you type int ( Á ) And if you type evalf(int ( Á )), you obtain , in agreement with the result near the beginning of the example Bessel Functions with Half-Integer Are Elementary We discover this remarkable fact as another property obtained from the series () and confirm it in the problem set by using Bessel s ODE EXAMPLE 3 Elementary Bessel Functions with The Value (, 3, 5, Á ) We first prove (Fig ) I x 3 (4x (x J J ) J ) 8 3J () J () J ()4 38J () 4J () J ()4 8 J () 4 J () 7J () 4J () J J () (a) J > (x) B px sin x, (b) J >(x) cos x B px The series () with is J > (x) x a The denominator can be written as a product AB, where (use (6) in B) here we used (proof below) m () m x m m> m! (m 3 ) B x a m () m x m m m! (m 3 ) A m m! m(m )(m 4) Á 4 #, B m (m 3 ) m (m )(m ) Á 3 # ( ) (m )(m ) Á 3 # # p ; (3) ( ) p The product of the right sides of A and B can be written Hence AB (m )m (m ) Á 3 # # p (m )!p J > (x) B px a m () m x m (m )! sin x B px π 4π 6π x Fig Bessel functions J > and J >

8 94 CHAP 5 Series Solutions of ODEs Special Functions This proves (a) Differentiation and the use of (a) with now gives This proves (b) From () follow further formulas successively by (c), used as in Example We finally prove ( ) p by a standard trick worth remembering In (5) we set t u Then dt u du and a b e t t > dt e u du [x J > (x)]r B p cos x x > J > (x) We square on both sides, write v instead of u in the second integral, and then write the product of the integrals as a double integral: a b 4 e u du e v dv 4 e (u v ) du dv We now use polar coordinates r, u by setting u r cos u, v r sin u Then the element of area is du dv r dr du and we have to integrate over r from to and over u from to p> (that is, over the first quadrant of the uv-plane): a p> b 4 e r r dr du 4 # p e r r dr a b er ` p By taking the square root on both sides we obtain (3) General Solution Linear Dependence For a general solution of Bessel s equation () in addition to J we need a second linearly independent solution For not an integer this is easy Replacing by in (), we have (4) J (x) x () m x m a m m m! (m ) Since Bessel s equation involves, the functions J and J are solutions of the equation for the same If is not an integer, they are linearly independent, because the first terms in () and in (4) are finite nonzero multiples of x and x Thus, if is not an integer, a general solution of Bessel s equation for all x is y(x) c J (x) c J (x) This cannot be the general solution for an integer n because, in that case, we have linear dependence It can be seen that the first terms in () and (4) are finite nonzero multiples of x and x, respectively This means that, for any integer n, we have linear dependence because (5) J n (x) () n J n (x) (n,, Á )

9 SEC 54 Bessel s Equation Bessel Functions J (x) 95 PROOF To prove (5), we use (4) and let approach a positive integer n Then the gamma function in the coefficients of the first n terms becomes infinite (see Fig 553 in App A3), the coefficients become zero, and the summation starts with m n Since in this case (m n ) (m n)! by (8), we obtain (6) () m x m n J n (x) a mn mn m! (m n)! a s () ns x sn sn (n s)! s! (m n s) The last series represents () n J n (x), as you can see from () with m replaced by s This completes the proof The difficulty caused by (5) will be overcome in the next section by introducing further Bessel functions, called of the second kind and denoted by Y PROBLEM SET 54 Convergence Show that the series () converges for all x Why is the convergence very rapid? ODES REDUCIBLE TO BESSEL S ODE This is just a sample of such ODEs; some more follow in the next problem set Find a general solution in terms of J and J or indicate when this is not possible Use the indicated substitutions Show the details of your work Two-parameter ODE x ys xyr (l x )y x ys xyr (x 49)y 4 xys yr 4 y (x z) ys (e x 9)y (e x z) (lx z) x ys 4 (x 3 4) y (y ux, x z) x ys xyr 4 (x )y (x z) (x ) ys (x )yr 6x(x )y (x z) xys ( )yr xy (y x u) x ys ( )xyr (x )y (y x u, x z) CAS EXPERIMENT Change of Coefficient Find and graph (on common axes) the solutions of ys kx yr y, y(), yr(), for k,,, Á, (or as far as you get useful graphs) For what k do you get elementary functions? Why? Try for noninteger k, particularly between and, to see the continuous change of the curve Describe the change of the location of the zeros and of the extrema as k increases from Can you interpret the ODE as a model in mechanics, thereby explaining your observations? CAS EXPERIMENT Bessel Functions for Large x (a) Graph J for n, Á n (x), 5 on common axes (b) Experiment with (4) for integer n Using graphs, find out from which x x n on the curves of () and (4) practically coincide How does x n change with n? (c) What happens in (b) if n? (Our usual notation in this case would be ) (d) How does the error of (4) behave as a function of x for fixed n? [Error exact value minus approximation (4)] (e) Show from the graphs that J (x) has extrema where J (x) Which formula proves this? Find further relations between zeros and extrema 3 5 ZEROS of Bessel functions play a key role in modeling (eg of vibrations; see Sec 9) 3 Interlacing of zeros Using () and Rolle s theorem, show that between any two consecutive positive zeros of J n (x) there is precisely one zero of J n (x) 4 Zeros Compute the first four positive zeros of J (x) and J (x) from (4) Determine the error and comment 5 Interlacing of zeros Using () and Rolle s theorem, show that between any two consecutive zeros of J (x) there is precisely one zero of J (x) 6 8 HALF-INTEGER PARAMETER: APPROACH BY THE ODE 6 Elimination of first derivative Show that y uv with v(x) exp ( p(x) dx) gives from the ODE ys p(x)yr q(x)y the ODE us 3q(x) 4 p(x) pr(x)4 u, not containing the first derivative of u

10 96 CHAP 5 Series Solutions of ODEs Special Functions 7 Bessel s equation Show that for () the substitution in Prob 6 is y ux > and gives (7) x u (x _ 4 )u 8 Elementary Bessel functions Derive () in Example 3 from (7) 9 5 APPLICATION OF (): DERIVATIVES, INTEGRALS Use the powerful formulas () to do Probs 9 5 Show the details of your work 9 Derivatives Show that Jr (x) J (x), Jr (x) J (x) J (x)>x, Jr (x) [J (x) J 3 (x)] Bessel s equation Derive () from () Basic integral formula Show that x J (x) dx x J (x) c Basic integral formulas Show that x J (x) dx x J (x) c, J (x) dx J (x) dx J (x) 3 Integration Show that x J (x) dx x J (x) xj (x) J (x) dx (The last integral is nonelementary; tables exist, eg, in Ref [A3] in App ) 4 Integration Evaluate x J 4 (x) dx 5 Integration Evaluate J 5 (x) dx 55 Bessel Functions Y (x) General Solution To obtain a general solution of Bessel s equation (), Sec 54, for any, we now introduce Bessel functions of the second kind Y (x), beginning with the case n When n, Bessel s equation can be written (divide by x) () xys yr xy Then the indicial equation (4) in Sec 54 has a double root r This is Case in Sec 53 In this case we first have only one solution, J (x) From (8) in Sec 53 we see that the desired second solution must be of the form () We substitute y and its derivatives into () Then the sum of the three logarithmic terms x Js ln x, Jr ln x, and x J ln x is zero because J is a solution of () The terms J >x and J >x (from xys and yr) cancel Hence we are left with Jr a m n y (x) J (x) ln x a yr Jr ln x J x a ma m x m ys Js Jr ln x x J x a m(m ) A m x m a m m m m A m x m m (m ) A m x m m A m x m a m A m x m

11 SEC 55 Bessel Functions Y (x) General Solution 97 Addition of the first and second series gives m A m x m The power series of Jr (x) is obtained from () in Sec 54 and the use of m!>m (m )! in the form () m mx m Jr (x) a m Together with m A and A m x m m x m this gives m (m!) a m () m x m m m! (m )! (3*) a m () m x m m m! (m )! a m m A m x m a A m x m m First, we show that the A with odd subscripts are all zero The power x m occurs only in the second series, with coefficient A Hence A Next, we consider the even powers x s The first series contains none In the second series, m s gives the term (s ) A s x s In the third series, m s Hence by equating the sum of the coefficients of to zero we have x s (s ) A s,, Á s A s, Since A, we thus obtain A 3, A 5, Á, successively We now equate the sum of the coefficients of to zero For s this gives x s 4A, thus A 4 For the other values of s we have in the first series in (3*) m s, hence m s, in the second m s, and in the third m s We thus obtain () s s (s )! s! (s ) A s A s For s this yields 8 6A 4 A, thus A and in general (3) A m,, Á m ()m m (m!) a 3 Á m b, Using the short notations (4) h h m Á m m, 3, Á and inserting (4) and A A 3 Á into (), we obtain the result y (x) J (x) ln x a m () m h m m (m!) x m (5) J (x) ln x 4 x 3 8 x4 3,84 x 6 Á

12 98 CHAP 5 Series Solutions of ODEs Special Functions Since J and y are linearly independent functions, they form a basis of () for x Of course, another basis is obtained if we replace y by an independent particular solution of the form a( y bj ), where a ( ) and b are constants It is customary to choose a >p and b g ln, where the number g Á is the so-called Euler constant, which is defined as the limit of Á s ln s as s approaches infinity The standard particular solution thus obtained is called the Bessel function of the second kind of order zero (Fig ) or Neumann s function of order zero and is denoted by Y (x) Thus [see (4)] (6) Y (x) p c J (x) aln x gb a m () m h m m (m!) x m d For small x the function Y (x) behaves about like ln x (see Fig, why?), and Y (x) : as x : Bessel Functions of the Second Kind Y n (x) For n,, Á a second solution can be obtained by manipulations similar to those for n, starting from (), Sec 54 It turns out that in these cases the solution also contains a logarithmic term The situation is not yet completely satisfactory, because the second solution is defined differently, depending on whether the order is an integer or not To provide uniformity of formalism, it is desirable to adopt a form of the second solution that is valid for all values of the order For this reason we introduce a standard second solution Y (x) defined for all by the formula (7) (a) (b) Y (x) sin p [J (x) cos p J (x)] Y n (x) lim :n Y (x) This function is called the Bessel function of the second kind of order or Neumann s function 7 of order Figure shows Y (x) and Y (x) Let us show that J and Y are indeed linearly independent for all (and x ) For noninteger order, the function Y (x) is evidently a solution of Bessel s equation because J (x) and J (x) are solutions of that equation Since for those the solutions and are linearly independent and involves, the functions and are J J Y J J Y 7 CARL NEUMANN (83 95), German mathematician and physicist His work on potential theory using integer equation methods inspired VITO VOLTERRA (8 94) of Rome, ERIK IVAR FREDHOLM (866 97) of Stockholm, and DAVID HILBERT (96 943) of Göttingen (see the footnote in Sec 79) to develop the field of integral equations For details see Birkhoff, G and E Kreyszig, The Establishment of Functional Analysis, Historia Mathematica (984), pp 58 3 The solutions Y (x) are sometimes denoted by N (x); in Ref [A3] they are called Weber s functions; Euler s constant in (6) is often denoted by C or ln g

13 SEC 55 Bessel Functions Y (x) General Solution 99 5 Y Y 5 x 5 Fig Bessel functions of the second kind Y and Y (For a small table, see App 5) linearly independent Furthermore, it can be shown that the limit in (7b) exists and Y n is a solution of Bessel s equation for integer order; see Ref [A3] in App We shall see that the series development of Y n (x) contains a logarithmic term Hence J n (x) and Y n (x) are linearly independent solutions of Bessel s equation The series development of Y n (x) can be obtained if we insert the series () in Sec 54 and () in this section for J (x) and J (x) into (7a) and then let approach n; for details see Ref [A3] The result is (8) Y n (x) p J n(x) aln x gb xn p a x n n p a (n m )! mn x m m! m m () m (h m h mn ) mn m! (m n)! x m where x, n,, Á, and [as in (4)] h, h, h m Á m, h mn Á m n For n the last sum in (8) is to be replaced by [giving agreement with (6)] Furthermore, it can be shown that Y n (x) () n Y n (x) Our main result may now be formulated as follows THEOREM General Solution of Bessel s Equation A general solution of Bessel s equation for all values of (and x ) is (9) y(x) C J (x) C Y (x) We finally mention that there is a practical need for solutions of Bessel s equation that are complex for real values of x For this purpose the solutions () H () (x) J (x) iy (x) H () (x) J (x) iy (x)

14 CHAP 5 Series Solutions of ODEs Special Functions are frequently used These linearly independent functions are called Bessel functions of the third kind of order or first and second Hankel functions 8 of order This finishes our discussion on Bessel functions, except for their orthogonality, which we explain in Sec 6 Applications to vibrations follow in Sec PROBLEM SET 55 9 FURTHER ODE s REDUCIBLE TO BESSEL S ODE Find a general solution in terms of J and Y Indicate whether you could also use J instead of Y Use the indicated substitution Show the details of your work x ys xyr (x 6) y xys 5yr xy ( y u>x ) 9x ys 9xyr (36x 4 6)y (x z) ys xy ( y ux, 3x 3> z) 4xys 4yr y (x z) xys yr 36y (x z) ys k x y (y ux, kx z) ys k x 4 y (y ux, 3 kx 3 z) xys 5yr xy (y x 3 u) CAS EXPERIMENT Bessel Functions for Large x It can be shown that for large x, () Y n (x) >(px) sin (x np 4 p) with defined as in (4) of Sec 54 (a) Graph Y for n, Á n (x), 5 on common axes Are there relations between zeros of one function and extrema of another? For what functions? (b) Find out from graphs from which x x n on the curves of (8) and () (both obtained from your CAS) practically coincide How does x n change with n? (c) Calculate the first ten zeros x m, m, Á,, of Y (x) from your CAS and from () How does the error behave as m increases? (d) Do (c) for Y (x) and Y (x) How do the errors compare to those in (c)? 5 HANKEL AND MODIFIED BESSEL FUNCTIONS Hankel functions Show that the Hankel functions () form a basis of solutions of Bessel s equation for any Modified Bessel functions of the first kind of order are defined by I (x) i J (ix), i Show that I satisfies the ODE () x ys xyr (x ) y 3 Modified Bessel functions Show that I (x) has the representation x m (3) I (x) a m m m! (m ) 4 Reality of I Show that I (x) is real for all real x (and real ), I (x) for all real x, and I n (x) I n (x), where n is any integer 5 Modified Bessel functions of the third kind (sometimes called of the second kind) are defined by the formula (4) below Show that they satisfy the ODE () (4) K (x) p sin p 3I (x) I (x)4 CHAPTER 5 REVIEW QUESTIONS AND PROBLEMS Why are we looking for power series solutions of ODEs? What is the difference between the two methods in this chapter? Why do we need two methods? 3 What is the indicial equation? Why is it needed? 4 List the three cases of the Frobenius method, and give examples of your own 5 Write down the most important ODEs in this chapter from memory 6 Can a power series solution reduce to a polynomial? When? Why is this important? 7 What is the hypergeometric equation? Where does the name come from? 8 List some properties of the Legendre polynomials 9 Why did we introduce two kinds of Bessel functions? Can a Bessel function reduce to an elementary function? When? 8 HERMANN HANKEL ( ), German mathematician

15 Summary of Chapter 5 POWER SERIES METHOD OR FROBENIUS METHOD Find a basis of solutions Try to identify the series as expansions of known functions Show the details of your work ys 4y xys ( x) yr (x ) y 3 (x ) ys (x ) yr 35y (x ) ys 3y x ys xyr (x 5) y x ys x 3 yr (x ) y xys (x ) yr y xys 3yr 4x 3 y 9 ys 4x y xys yr xy SUMMARY OF CHAPTER 5 Series Solution of ODEs Special Functions The power series method gives solutions of linear ODEs () ys p(x) yr q(x)y with variable coefficients p and q in the form of a power series (with any center x, eg, x ) () y(x) a a m (x x ) m a a (x x ) a (x x ) Á Such a solution is obtained by substituting () and its derivatives into () This gives a recurrence formula for the coefficients You may program this formula (or even obtain and graph the whole solution) on your CAS If p and q are analytic at x (that is, representable by a power series in powers of x x with positive radius of convergence; Sec 5), then () has solutions of this form () The same holds if h, p, q in are analytic at x and so that we can divide by h (x ), h and obtain the standard form () Legendre s equation is solved by the power series method in Sec 5 The Frobenius method (Sec 53) extends the power series method to ODEs (3) m h (x)ys p(x)yr q (x)y ys a(x) yr b(x) x x (x x ) y whose coefficients are singular (ie, not analytic) at x, but are not too bad, namely, such that a and b are analytic at x Then (3) has at least one solution of the form (4) y(x) (x x ) r a a m (x x ) m a (x x ) r a (x x ) r Á m