Sturm-liouville problem

(See attached file for full problem description) --- Use the following table to solve 3 and 4. J0(x) J1(x) Y0(x) Y1(x) 2.4048 0.0000 0.8936 2.1971 5.5201 3.8317 3.9577 5.4297 8.6537 7.0156 7.0861 8.5960 11.7915 10.1735 10.2223 11.7492 14.9309 13.3237 13.3611 14.8974 3. Find the first four α i 0 defined by J1( 3α ) = 0 4. Find the first four α i 0 defined by J'0( 2α ) = 0.

Bessel and Legendre series

(See attached file for full problem description) --- 8. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = 1/2(3x2- 1). If x = cosθ , then P0( cosθ ) = 1 and P1( cosθ ) = cos θ . Show that P2( cosθ ) = 1/4( 3cos2θ + 1 ). 9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (θ) = )of F( θ ) = 1 - cos2θ . 10. Why is a Fourier-Legendre expansion of a polynomial function that is defined on the interval ( -1, 1 ) necessarily a finite series. 11. Using only your conclusions from problem 10, find the finite Fourier-Legendre series of f(x) = x2 ---

Stochastic Differential Equations.

(See attached file for full problem description)

2 Dimensional wave equation

Please derive the TWO Dimensional wave equation. Note: derive this equation in such a way a beginner in PDE will understand. (put comment and or explain where needed)

Stochastic DE (density function,E(x).

I know this problem is very easy, finding E(x) from the distribution function. I tried to do it by integration by parts, one way I took x to be my first function and the whole exp term to be my 2nd function, but it didn't work, then I split the exponential term to 2 terms combined one with x as my first function and took the second as the 2nd function but it didn't work either. Can someone help me? Please provide a detailed proof/answer. Justify all your claims. See attached file for full problem description.