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Chapter 1: Q 24E (page 1)

Question:Use a CAS to graphJ3/2(x),J-3/2(x),J5/2(x), and J-5/2(x).

Short Answer

Expert verified

The graph has been plotted.

Step by step solution

01

Step 1:Define Spherical Bessel’s equation.

Bessel functions of half-integral order are used to dene two more important functions:

jn(x)=蟺2xJn+1/2(x)

yn(x)=蟺2xYn+1/2(x)

The function jn (x)is called the spherical Bessel function of the first kind and yn (x) is the spherical Bessel function of the second kind.

02

Find the graph of j3/2 (x) .

Use GNU Octave to plot the functions.

03

Find the value of j-3/2(x) .

Let,

04

Step 4:Find the value of j5/2 (x) .

Let,

05

Step 5:Find the value of j-5/2 (x) .

Let,

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Most popular questions from this chapter

Question: The Taylor series for f(x) =ln (x)about x2=0given in equation (13) can also be obtained as follows:

(a)Starting with the expansion 1/ (1-s) =鈭�n=0鈭�s'' and observing that

'

obtain the Taylor series for 1/xabout x0= 1.

(b)Since use the result of part (a) and termwise integration to obtain the Taylor series for f (x)=lnxaboutx0= 1.

(a) Show that y2+x-3=0 is an implicit solution todydx=-12y on the interval (-鈭�,3).

(b) Show thatxy3-xy3sinx=1 is an implicit solution todydx=xcosx+sinx-1y3x-xsinx on the interval (0,蟺2).

In Problems 9鈥�13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

exy+y=x-1,dydx=e-xy-ye-xy+x

Competing Species. Let pi(t) denote, respectively, the populations of three competing species Si,i=1,2,3.Suppose these species have the same growth rates, and the maximum population that the habitat can support is the same for each species. (We assume it to be one unit.) Also, suppose the competitive advantage thatS1 has overS2 is the same as that ofS2 overS3 and over. This situation is modeled by the system

p'1=p1(1-p1-ap2-bp3)p'2=p2(1-bp1-p2-ap3)p'3=p3(1-ap1-bp2-p3)

where a and b are positive constants. To demonstrate the population dynamics of this system when a = b = 0.5, use the Runge鈥揔utta algorithm for systems with h = 0.1 to approximate the populations over the time interval [0, 10] under each of the following initial conditions:

(a)p1(0)=1.0,p2=0.1,p3=0.1(b)p1(0)=0.1,p2=1.0,p3=0.1(c)p1(0)=0.1,p2=0.1,p3=1.0

Decide whether the statement made is True or False. The function y(x)=-13(x+1) is a solution to dydx=y-1x+3.
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