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).
$xy'' + 5 y + xy = 0$

Short Answer

Expert verified

The solution of the differential equation $xy'' + 5 y + xy = 0 is y = x^{- 2} [{AJ}_{2} (x) + {BN}_{2} (x)]$ .

Step by step solution

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

Consider the standard form is,

$y' + \frac{1 - 2 a}{x} y' + [{({bcx}^{c - 1})}^{2} + \frac{a^{2} - P^{2} c^{2}}{x^{2}}] y = 0$

Solution to the above mentioned standard equation is,

$y = x^{a} Z_{p} ({bx}^{c})$

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

Consider the given equation as follow.

$xy'' + 5 y' + xy = 0$

Rearrange the above equation as follow.

$y'' + \frac{5 y'}{x} + y = 0$

Compare the given equation with standard differential equation as follows:

$1 - 2 a = 5 . . . . (1) {(bc)}^{2} = 1 . . . . (2) 2 (c - 1) = 0 . . . . . (3) a^{2} - p^{2} c^{2} = 0 . . . . . . (4)$

Therefore, the values are as follows:

$- 2 a = 5 - 1 a = - \frac{4}{2} a = - 2$

By solving equation (1), you have

$(c - 1) = 0 c = 1$

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

${(- 2)}^{2} - p^{2} {(1)}^{2} = 0 4 - p^{2} = 0 p^{2} = 4 p = 2$

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

b = 1

Hence, the solution is

$y = x^{- 2} Z_{2} (x)$

Therefore, $y = x^{- 2} [{AJ}_{2} (x) + {BN}_{2} (x)]$ .

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Find the solutions of the following differential equations in terms of Bessel functions by using equations (16.1) and (16.2).
$xy'' + 5 y + xy = 0$

Short Answer

Step by step solution

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

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

One App. One Place for Learning.

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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

Step by step solution

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

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Find the solutions of the following differential equations in terms of Bessel functions by using equations (16.1) and (16.2).
$xy'' + 5 y + xy = 0$