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

Astrophysics

Electricity

Radiation

Translational Dynamics

Fields in Physics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

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