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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

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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)=π2xJn+1/2(x)

yn(x)=π2xYn+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

The temperatureT(in units of 100 F) of a university classroom on a cold winter day varies with timet(in hours) asdTdt=1-T,ifheatingunitsisOn-T,ifheatingunitsisOFF.

T=0Suppose at 9:00 a.m., the heating unit is ON from 9-10 a.m., OFF from 10-11 a.m., ON again from 11 a.m.–noon, and so on for the rest of the day. How warm will the classroom be at noon? At 5:00 p.m.?

Show thatϕx=c1sinx+c2cosx, is a solution tod2ydx2+y=0 for any choice of the constantsc1andc2. Thus,c1sinx+c2cosx, is a two-parameter family of solutions to the differential equation.

Consider the question of Example 5 ydydx-4x=0

  1. Does Theorem 1 imply the existence of a unique solution to (13) that satisfiesy(x0)=0?
  2. Show that when x0≠0equation (13) can’t possibly have a solution in a neighbourhood of x=x0that satisfies y(x0)=0.
  3. Show two distinct solutions to (13) satisfying y(0)=0 ( See Figure 1.4 on page 9).

In problems 1-4 Use Euler’s method to approximate the solution to the given initial value problem at the points x=0.1,0.2,0.3,0.4, and 0.5, using steps of size 0.1h=0.1.

dydx=-xy,y(0)=4

Variation of Parameters. Here is another procedure for solving linear equations that is particularly useful for higher-order linear equations. This method is called variation of parameters. It is based on the idea that just by knowing the form of the solution, we can substitute into the given equation and solve for any unknowns. Here we illustrate the method for first-order equations (see Sections 4.6 and 6.4 for the generalization to higher-order equations).

(a) Show that the general solution to (20)dydx+P(x)y=Q(x) has the formy(x)=Cyh(x)+yp(x),whereyh ( ≠0is a solution to equation (20) when Q(x)=0,

C is a constant, andyp(x)=v(x)yh(x) for a suitable function v(x). [Hint: Show that we can takeyh=μ-1(x) and then use equation (8).] We can in fact determine the unknown function yhby solving a separable equation. Then direct substitution of vyh in the original equation will give a simple equation that can be solved for v.

Use this procedure to find the general solution to (21) localid="1663920708127" dydx+3xy=x2, x > 0 by completing the following steps:

(b) Find a nontrivial solutionyh to the separable equation (22) localid="1663920724944" dydx+3xy=0, localid="1663920736626" x>0.

(c) Assuming (21) has a solution of the formlocalid="1663920777078" yp(x)=v(x)yh(x) , substitute this into equation (21), and simplify to obtain localid="1663920789271" v'(x)=x2yh(x).

d) Now integrate to getlocalid="1663920800433" vx

(e) Verify thatlocalid="1663920811828" y(x)=Cyh(x)+v(x)yh(x) is a general solution to (21).
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