5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variab...
6. Express solutions of the following in terms of the special functions defined in lectures [do not derive these solutions]: (a) (1-2)y" - 2ry n(n+ 1)y 0, -1S1, write down the solution that satisfies the boundary conditions y(-1) = (-1)", y(1) = 1; to sin θ + n (n + 1)7-0 0 θ π, write down the general solution and then the solution sin 0 dø Sin 0 that is bounded forve e [0, π] and 2r-periodic with respect to...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: 16k2 1 16k2 1 (16k2 1)2 に1...
2. Consider the function f(x) defined on 0 <x < 2 (see graph (a) Graph the extension of f(x) on the interval (-6,6) that fix) represents the pointwise convergence of the Sine series. At jump discontinuities, identify the value to which the series converges (b) Derive a general expression for the coefficients in the Fourier Sine series for f(x). Then write out the Fourier series through the first four nonzero terms. Expressions involving sin(nt/2) and cos(nt/2) must be evaluated as...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck. (4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
Consider the function f defined on the interval-5, 5 as follows, { E5,0), те (0,5). 3. f(x) = 3. Denote by fr the Fourier series expansion of fon [-5,5]. fF(x)= 2 +b sin а, cos Find the coefficients a, an and b with n> 1. 0 an b = (10-10(-1)^n)/(pin) M M M Consider the function f defined on the interval-5, 5 as follows, { E5,0), те (0,5). 3. f(x) = 3. Denote by fr the Fourier series expansion of...
Solve the equation on the interval [0.2π). (tan θ + 1)( cos θ-1):0 Use an identity to solve the following equation on the 2 2 cos x sin x 1 0 Select the correct choice below and, if necessary, filli (Type your answer in radians. Use integers ort separate answers as needed.) There is no solution. 0 B. Solve the equation on the interval [0.2π). (tan θ + 1)( cos θ-1):0 Use an identity to solve the following equation on...
Please explain the solution and write clearly for nu, ber 25. Thanks. 25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
Solve the following equation on the interval [O,2π). cos x + 2 sin x cos x = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. ●A (Type an exact answer, using π as needed. Use a comma to separate answers as needed. T O B. There is no solution. Solve the equation on the interval [0,2 2 cot x = cot' x sin x Select the correct choice below and,...
2. In the lecture the general solution to the Legendre equation (1-z?)y', _ 2 ry, + n(n + 1)У-0.TIER. х є R series u(x) and ():(r) don(r) + of convergence of y1 (a), y2(z) considering: (i) the paraneter n is nonnegative înteger, n є N; (ii) the parameter n is not an integer, n ¢ Z. [Do not derive these series, refer to the relevant results obtained in lecture] 2. In the lecture the general solution to the Legendre equation...
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy = 0 (1) where a, b, c are constants satisfy so that y(x) = x (a) Find and justify what conditions should a constant m to (1) is a solution (b) Using your solution to (1) Write these three different cases as an equation that a, b,c satisfy. Hint: Use the quadratic formula we should get three different cases for the values that m can...