MATC46 Winter 2019
Assignment 5
Due Date: Monday 25 March.
Problems:
1. Prove that 2.
2. Determine the order p of the following Bessel’s equations. Find the general solution for
each of the equation and write down three terms of the Bessel function of the first kind of
p.a) 2 2
x y xy x y ′′ ′ ++ = ( 9) 0 b) 2 2 1 ( )0 4 x y xy x y ′′ ′ ++ =
3. Show that 1 .
4. Verify that y1 = xp
Jp(x) and y2 = xp
Yp(x) are linearly independent solutions of
xy p y xy ′′ ′ + + = (1 2 ) 0, x > 0 .
5. Evaluatea) 4
3 () x J x dx ∫ b) 1 () J x dx ∫ c) 2
3 () x J x dx ∫ d) 3
0 () x J x dx ∫ .
6. Approximate the given function by a Bessel series of the given p.
p = 1 b) f(x) = J0(x); 0 < x < α01, p = 0.
7. Solve the vibrating membrane problem (symmetric case)
a) a = 1, c = 1, f(r) = J0(α1r), g(r) = 0. b) a = 1, c = 1, f(r) = J0(α3r), g(r) = 1 – r2.
8. Using separation pf variables to find the solution of heat boundary value problem
What is the solution if f(r) = 100, 0 < r < a?
9. Solve the vibrating membrane problemcos2θ, g(r, θ) = 0, a = 3, c = 1.
b) F(r, θ) = 0, g(r, θ) = (1 – r2)r2
sin2θ, a = c = 1.
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