For the PDE in Problem 29, assume the following boundary conditions are imposed:
Show that a non-trivial solution of the form
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- 4. Consider a square service region of unit area in which travel is right angle and directions of travel are parallel to the sides of the square. Let (X, Y₁) be the location of a mobile unit and (X₂, Y₂) the location of a demand for service. The travel distance is D =Dx + Dy where Dx = |X₁ - X₂ and Dy = |Y₁ — Y2\. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for Dx and Dy is (4(1-x)(1-y), fpx.D¸y (x, y) = {4(1 – b. Define Ryx = D/Dr. Show that the pdf of Ryx is 2 3 fryx (r) = . 2 3r² 1 r, 3 1 3r3' 0, 0≤x≤ 1,0 ≤ y ≤ 1 otherwise 0 ≤r≤1 1 ≤r <∞0arrow_forwardBy using r -software please for parts f and garrow_forward. Solve the following problem with insulated boundaries: U = Uxx Ju,(0, t) = 0 lu(1, t) = 0 u(x,0) = f(x) P.D.E. 0< x <1 0arrow_forward4) Let f(x,y, z ) = . In(r+2y + 3z). Find Vf =arrow_forward1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, [=" dz = 1 -(z2+¹ - z+¹) ਬਾ n+1 (n = 0, 1, 2, ...).arrow_forward12. Suppose that f(x, y, z, w) = 0 and g(x, y, z, w) = 0 determine z and w as differentiable functions of the independent variables x and y, and suppose that af åg dz əw of dg # 0. dw dz Show that of åg af dg дz ax dw дw дх дх af dg af og дг дw dw dz and af dg дz ду of dg ду дг дw ду of dg af dg ме z0 dw dz ||arrow_forwardLet f: R² → R be defined by f(x, y) = 3x¹ + 7y³ +7 and set z = f(x, y). Find an equation for the contour line (level set) of f at level z= 9. help (equations)arrow_forward4. (a) Prove, without recourse to geometry, that u. (V x W) = v. (w xu) = w (u xv) = -u. (w x V) = −w. (V x u) = -v. (u x W).arrow_forward12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3Darrow_forward3) Evaluate ,(y+z)dx + (x + z)dy + (x +y)dz, C consists of line segments from (0,0,0) to (1,0,1), and from (1,0,1) to (0,1,2).arrow_forwardSolve the following P.D.E. , with the following conditions: u(0,y) = 0, u(L,y) = 0, u(x,0) = 0 and u(x,m) = x2 P.D.E a'u 0: ax? a y ||arrow_forwardLet the joint p.m.f. of X and Y be defined by 3x + y 45 f(x, y) = 3 x = 1, 2, 3 y = 1,2 a) Find fx (x), the marginal p.m.f. of X, and fy (y), the marginal p.m.f. of Y. b) Are X and Y independent? Please explain your reasoning. c) Find P(X+Y<4).arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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