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Chapter 12: Q12P (page 594)

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

xy''+5y+xy=0

Short Answer

Expert verified

The solution of the differential equation xy''+5y+xy=0isy=x-2AJ2x+BN2x .

Step by step solution

01

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

Consider the standard form is,

y'+1-2axy'+[bcxc-12+a2-P2c2x2]y=0

Solution to the above mentioned standard equation is,

y=xaZp(bxc)

02

Find the solutions of xy''+5y'+xy=0differential equations:

Consider the given equation as follow.

xy''+5y'+xy=0

Rearrange the above equation as follow.

y''+5y'x+y=0

Compare the given equation with standard differential equation as follows:

1-2a=5....(1)(bc)2=1....(2)2(c-1)=0.....(3)a2-p2c2=0......(4)

Therefore, the values are as follows:

-2a=5-1a=-42a=-2

By solving equation (1), you have

c-1=0c=1

Substitute -2 for a and 1 for c into equation (4).

-22-p212=04-p2=0p2=4p=2

And putting 1 for c into equation (2) you get,

b = 1

Hence, the solution is

y=x-2Z2(x)

Therefore, y=x-2AJ2x+BN2x.

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Most popular questions from this chapter

Computer plot on the same axes several IP(X) functions together with their common asymptotic approximation. Then computer plots each function with its small X approximation.


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.]

To study the approximations in the table, a computer plot on the same axes the given function together with its small approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agreeing with the function for large . If the small approximation is not clear, plot it alone with the function over a small interval y2(x).

Show thatJP(x)N'P(x)-J'P(x)NP(x)=J'P(x)J-p(x)-Jp(x)j'-p(x)蝉颈苍辫蟺=2蟺虫.

Expand the following functions in Legendre series.
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