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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)] =  then L[J1 (t)]= _____

(a)

(b)

(c)

(d)

Explanation: L[J0(t)] =

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 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)

Explanation: xy’’ + ay’ + k2xy = 0;

Question 3: The value of is

(a)

(b)

(c)

(d) 0

Explanation: For a>0

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

(a) 1

(b)

(c)

(d)

Explanation: Bessel function of trigonometric function 2 = sin x

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]

Explanation: The trigonometric expansion of sin x

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

(a)

(b) 1

(c) 0

(d)

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

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)]

Explanation: xy’’ + ay’ + k2yx = 0

Question 8: The general solution of the DE y’’ -y’ + 4(x2- )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)]

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

Question 9:  dx is equal to

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

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

(c) x3J1(x)- 2x2J2(x)

(d) None

Explanation:

Question 10: The general solution to the DE  x2y’’+ xy’+(4x2- )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)