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Bessel Function

Bessel functions are mathematical functions. It was defined by Daniel Bernoulli. Later it was generalized by Friedrich Bessel. It is a cylinder function defined in 1817.

 

Bessel Function Sample Questions:

Question 1: Let J0(.) and J1(.) be the Bessel function of the first order kind of order zero and one respectively. If L[J0(t)] =  Bessel function of the first order kind of order zero then L[J1 (t)]= _____


(a) option a

(b) option b

(c) option c

(d) option d


Answer: (c)

Explanation: L[J0(t)] =  explanation 1

 

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Question 2: If Jn(x) and Yn(x) denote Bessel functions of order n of the first and second kind, then general l solution of the DE then general l solution of the DE is given by


(a) y(x) = αx J1(x) +βxy1(x)

(b) y(x) = α J1(x) +βy1(x)

(c) y(x) = α J0(x) +βy0(x) 

(d) y(x) = αx J0(x) +βxy0(x)


Answer: (a)

Explanation: xy’’ + ay’ + k2xy = 0;

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Question 3: The value of The value of is


(a) option 1

(b) option 2

(c) option 3

(d) 0


Answer: (c)

Explanation: For a>0 For a>0

 

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Question 4: It is known that Bessel’s function Jn(x), n>0 Bessel’s function Jn(x)  = J0(x) +   n>0for all t>0, x∈R, then the value of J0(x)is equal to ___when x = find the value of

 

(a) 1

(b) question 3 option a

(c) question 3 option b

(d) question 3 option c


Answer: -2

Explanation: Bessel function of trigonometric function 2Bessel function of trigonometric function = sin x 

 

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Question 5: Trigonometric expansion of sin x involving Bessel function is

 

(a) 2[J1  - J3 + J5….]

(b) J0 - 2J4 +2J6 - 2J8

(c) J1 - 2J3 +2J5 - 2J7

(d) ) 2[J0 - J4 +J6 - J8]


Answer: (a)

Explanation: The trigonometric expansion of sin x

 

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Question 6:  It is known that Bessel’s function Jn(x), n>0 known that Bessel’s function Jn(x), n>0 =  J0(x) + J(x) for all z>0 and x∈ R, then the value of
then the value of is equal to

 

(a) Question 6 optio a

(b) 1

(c) 0

(d) Question 6 optio d


Answer: (d)

Explanation: Bessel function of trigonometric function J0(x) +Bessel function of trigonometric function J(x)= cos(x)

 

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Question 7: The general solution of the DE xy’’ -3y’ +xy =0 is


(a) x2[C1J1(x) + C2J-1(x)]

(b) x2[C1J2(x) + C2Y2(x)]

(c)  x[C1J2(x) + C2Y2(x)]

(d) x2[C1J2(x) + C2J-2(x)]


Answer: (b)

Explanation: xy’’ + ay’ + k2yx = 0 

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Question 8: The general solution of the DE y’’ -The general solutiony’ + 4(x2- find)y = 0 is

 

(a) x3/2[C1J5/4(x) + C2J-5/4(x)]

(b) x3/2[C1J5/4(x2) + C2J-5/4(x2)]

(c) x3/2[C1J5/4(x2) + C2Y5/4(x2)]

(d) x3/2[C1J5/4(x) + C2Y5/4(x2)]


Answer: (b)

Explanation: y’’ -multiplying the equation by x2y’ + 4(x2- multiplying the equation)y = 0 on multiplying the equation by x2

 

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Question 9: dx is equal to dx is equal to

 

(a) xJ0(x) - x3J1(x)

(b) x2J0(x)+ J1(x)

(c) x3J1(x)- 2x2J2(x)

(d) None

 

Answer: (c)

Explanation: solution of question 9

 

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Question 10: The general solution to the DE  x2y’’+ xy’+(4x2- The general solution to the DE )y = 0 in terms of Bessel’s functions JV(x) is

 

(a) y(x) = C1J3/5(2x) + C2J-3/5(2x)

(b) y(x) = C1J3/5(2x) + C2J-3/5(2x)

(c) y(x) = C1J3/5(x) + C2J-3/5(x)

(d) y(x) = C1J3/10(2x) + C2J3/10(2x)


Answer: (a)

Explanation: x2y’’+ xy’+(4x2-  )y = 0
 

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