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Chapter 12: Q5P (page 594)

Find the solutions of the following differential equations in terms of Bessel functions by using equations (16.1) and (16.2).

y''-1xy'+(4+1x2)y=0

Short Answer

Expert verified

The solution of the differential equation y''-1xy'+(4+1x2)y=0is

role="math" localid="1659265236089" y=xAJ0(2x)+BN0(2x).

Step by step solution

01

Concept of Bessel functions by using equations (16.1) and (16.2):

Consider the standard form as,

y''+1-2axy'+[bcxc-1+a2-p2c2x2]y=0

Solution to the above mentioned standard equation is,

y=xaZp(bcc)

02

Find the solutions of y''-1xy'+(4+1x2)y=0differential equations:

Consider the given equation as follow.

y''-1xy'+(4+1x2)y=0

Compare the given equation with standard differential equation as follows:

1-2a=-1(bc)2=42(c-1)=0a2-p2c2=1

Therefore, the given values are as follows:

a=16c=142p=13b=4

Hence, the solution is y=xZ0(2x).

Therefore, y=xAJ0(2x)+BN0(2x).

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Most popular questions from this chapter


Use the recursion relation (5.8a) and the values of P0(x)and P1(x) to find localid="1664340078504" P2(x)P3(x),P4(x),P5(x), and P6(x). [After you have found P3(x), use it to find P4(x) and so on for the higher order polynomials.]

As in Problem 1, study the Kp (X) functions. It is interesting to note (see Problem $17.4) that K1/2(X) is equal to the asymptotic approximation.

Expand the following functions in the Legendre series.

F(x)=[-1-1鈮�x010鈮�x1]

To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval j2(x)

We obtained (19.10) forJp(x),p鈮�0.It is, however, valid forp鈮�-1, that is forNp(x),0鈮�p<1. The difficulty in the proof occurs just after (19.7); we said that u,v,u',v'are finite at x=0which is not true forNp(x).

However, the negative powers of x cancel ifp<1. Show this for p=12by using two terms of the power series (12.9) or (13.1) for the function N1/2(x)=-J-1/2(x) [see (13.3)].
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