Let u = u(z. y. t) satisfy the following PDE and boundary conditions: &2U,, + uyy = 0. 0 < y < 1. 5 E R, t > 0, uy(z. 0, t) = 0, uy(z, 1. t) + &%Ltt(S. 1. t) = 0. z E R, t > 0, where E is small. Assuming a perturbation expansion u = uo(z.t) + U1(z,y.t)E2 + ... . show that uo satisfies the wave equation.

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A: It will serve as a joint Probability density function of a pair of continuous random variable x and…

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Q: 5. Suppose z = f (x, y), where x= g(s,t)and y= h(s,t). Show that ôz d²x ¸ ôz ô²y %3D ô°z ôx ôy ôz(…

A: Chain rule of partial derivatives: Let us consider a two variable function  z=f(x,y)  where ,…

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A: Here given that                          w=f(x,y,z)=ln(x2+y2+z2) ........................(1)…

Q: SupposeThat w=XYz+XY +Z+3andx=3cost, %3D Y=35int, z=27, Evaluete dwl dtt= I 2

A: solution.

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A: We need to evaluate the integral ∫Cydx+xdy where C is the line segment from (1,-1) to (4,3).

Q: 15) Consider the vector field F (x, y, 2) = (In y, x, z) and the oriented curve C:r (t) = (t, e', 2…

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A: X and Y are independent if their joint PMF is equal to the product of their marginal PDF .

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Q: 8. What is the relationship between a and b so that F(x, y) = (ary, 3ax + bry) is source free?

A: That's easy. Have a great day!!!

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Q: 2) Given the joint pmff(x.y): X 0 X 5 X 6 X -1 X 2 Y=-3 0.1 0.25 0 0.2 0 Y-1 0 0.1 0.05 0 0 0.1 0.05…

A: Given Joint pmf f(x,y) as shown belowThe marginal pmfs are calculated using the formulas shown below

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A: Two events, A and B, are independent ifP(A and B) = P(A) x P(B)We can then rearrange this…