Evaluate // e-(x+vº)dA where D = {(x,y): x2 + y2 <1,1 20, y 2 0}.
Find the infinite series solution to the following differential equations using method of Frobenius. 3xy"+(3-x)y'-y=0 at x = 0
2. Use the power series method to solve the following initial-value problem: y" + 2xy' + 8y = 0 with y(0) = 3 and y(0) = 0.
Use the method of Frobenius to obtain linearly independent series solutions about r = 0. 1.0"y" + 1ry' + (22 – 1)y=0. Use an initial index of k = 2 to develop the recurrence relation. The indicial roots are(in ascending order) rı = .12= Corresponding to the larger indicial root, the recurrence relation of the solution is given by C = Xq-2. The initial index is k = The solution is yı = (Q10 where Q1 = + Q222 +230...
Use power series y' = 1/(2 - x) y(0) = 2
Differential Eqs
Use the method of Frobenius to obtain one power series solution about x = 0: 2.
Use the method of Frobenius to obtain one power series solution about x = 0: 2.
(a) Suppose an is an infinite series such that 0 < Ant1 < an for all n. Either give an argument that such a series always converges, or else explain why this is not the case (by a general argument or an example). n=1 (b) The nth derivative of the function In(1 + x) is (-1)"- (n − 1)! (1 + x)" Use the Taylor remainder theorem to show that the polynomial p(x) = x - 3x2 + 323 will...
5. Use Newton's Binomial Theorem to write y- V1- 2 as an infinite series (write out at least four non-zero terms) and apply Rule I of De analysi term-by-term to find the area under the curve y-V1-x2 in the first quadrant (again using at least four non-zero terms). Using only the first four non-zero terms in the series you found, estimate the value of T. (Remark: This is a very inefficient way to estimate π. Using 1000 terms of the...
Use power series methods to solve the initial-value problem y''-2xy'+8y=0 y(0)=3 y'(0)=0 You must show your work and the power series method You only need to show the first four nonzero terms of each series in your answer
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise with zero mean and unit variance. (a) Is the model stationary? Explain your answer briefly. (b) Is the model invertible? Explain your answer briefly. (c) Find the infinite moving-average representation of Xt. Namely, find b; such that X =< 0;&–; j=0 (d) Evaluate the first three lags of the ACF and PACF.