Q4*) Find the extremals y, z of the the functional 1 = √² (2yz — 2z² + y² — z¹²) dx, - - with y(0) = 0, y(1) = 1, z(0) = 0, z(1) = 0.
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Q: Consider the weighted voting system [18: 15, 8, 3, 1]Find the Banzhaf power distribution of this…
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- 13 Find the linearizatan 2=L(x.y) f the function 2= {(x,y)=4x²-y²+2y at the point (x.y) = (3,2).The graph of y =x-3x-24xr+32 is concave upward on the interval (1, oo ) because: Oy' is positive on the interval (1, o0) Oy is negative on the interval (1, oo) Oy" is positive on the interval (1, o0) y" is negative on the interval (1, 00)Find the centroid of the thin plate bounded by the graphs of g(x)=x² and f(x) = -x + 56. Use the equations shown below with 8 = 1 and M = area of the region covered by the plate. b x = -1/1S5x[f(x) - Sx[f(x) - g(x)] dx M a b y = /[²(x) - g² (x)] dx The centroid of the thin plate is (x,y), where x = (Type integers or simplified fractions.) 1 a and y =
- Q.3) Find the constants a, b, c, d, e if f(z) = (ax³ + bxy² + 3x² + cy² + x) + i(dx²y – 2y³ + exy+ y) is analytic.Verify that the given function u is harmonic in an appropriate domain D. Findits harmonic conjugate v and find analytic function f(z) = u + iv satisfying theindicated condition.u(x, y) = 4xy^(3) − 4x^(3)y + x, f(1 + i) = 5 + 4i.Find all residues of the function 2z+1 z(2-2)' not including the residue at infinity.
- The function z(x, y) obeys, 8²u = -2e-² Ərəy Which of the follow represents a valid first step for finding u(x, y)? of-2e dx = -2e¯²y+C(y) of-2e²dy = -2eyf-2e dy = 2e " + C(y) of-2e-dy=-2e-2y+C(x) -2e-²dr = 2e-² Of-2e²dx = 2e-* + C(y) Which of the following represents the most general solution for u(x, y)? O 2e x + A(x) + B(y) O 2e-2y+ A(x) + B(y) O 2e-2y+A(x)B(y) ○ 2e yConsider the function. y = (2x² + 1)3 (a) Decompose the function in the form y = } {f(u), u} = dy = dx { dy (b) Find as a function of x. dx 12x (2x² + 1)² f(u) and u = g(x). (Use non-identity functions for f(u) and u.)