4. Check whether u(x, y) = y³ – 3x²y is harmonic or not. If u(x, y) is harmonic, then find its conjugate harmonic v(x, y).

Q: ду Let y = In Vx² + 2z3 + cos(b³), find

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Q: Q7 Show that CX,y) =e is Condinuous at co,0)

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A: Here given y=cos-1(z/x). We prove that zyy+z=0.

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A: 4. Given data is : w=xz+yz , x=cos2t , y=sin2t  and z=1t To express dwdt in terms of t

Q: 3. Find dy, where; a) cos(ry) - e 2 elx+y). b) ė?+ Ptany =4, u= xlax (use chain rule)

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Q: how that u(x, y) is harmonic in some domain and find a harmonic njugate v(x, y) hen: u(x, y) = 2x –…

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Q: Find Duf at P. 3 4 f (x, y) = sin(3x – 4y); P(4,3); u = i j %3D - - NOTE: Enter the exact answer.…

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Q: B. and , if z = (x² + y²)³/2, and r = e-" sin v, y = e¬u cos v. В. %3D

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Q: Eliminate the arbitrary constant/s of this equation to obtain the DE. Express the DE in general…

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Q: 1. If H(y) = cos y – sin y, Find H (0), H (),H (1T), and also show that %3D + x) = H (T – x) = -H…

A: Given that H(y)=cosy-siny find H(0), Hπ2, Hπ Show that  Hπ2+x=Hπ-x=-H-x

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Q: For the relation siny – y² =x²y– x+x , is dx 3x²y+1 А. cos y – 2y –x T – 1 3x² + 2y – cos y cos y –…

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Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

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Q: Calculate the “n” of the DE with the given 6y^2dx-x(2x^3+y)dy=0

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Q: 8:54 1 3. If y = (5x)sin(3x) , then %3D dy %3D dx Type here to search

A: This question is related to differentiations. Solve in 2nd step.

Q: Solve the Ro liowing B E 2 X + 2 -zey= zX

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Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) (7 Marks) When:…

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Q: Given: y = sqrt(u)  ,  u = 2x2 + 3x + 4 Determine dy/dx at x = -3 a) Using Leibniz Chain Rule b) By…

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Q: Given z(x, y) =In(2xy) where x(u,v)= e" cos(4v) and y(u,v)=e" sin(4v). Find ди 1 cos (4r)+-sin(4v)…

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Q: Verify that the indicătéd function is an exp for each solution. y" - 6y' + 25y = 0; y = e3x cos 4x…

A: Recall: (uv)'=u'v+uv' where u and v are the function of x

Q: Q3/ Find the complex integral 3+1 z2 dz 0+01 Along the path 1 y =,x 3*

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Q: 8. Verify that the function u(x, y) = eª sin(y) is harmonic in the region where it is defined and…

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Q: 5. If z = f(x, y), where x = r cos 0 and r sin 0, find z and and hence show that: (). (-(( 2 2 dz dz…

A: To find the partial derivatives of the following function in two variables by chain rule.

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Q: III 6(x3 + xy²) dv, where E is the solid in the first octant that lies beneath the paraboloid z = 4…

A: Given,       ∬∫E6(x3+xy2)dV,       z=4-x2-y2

Q: sin(iny)dy is equal to * y (A none of these cos(ny) + C y cos(Iny) + C (D) -cos(iny) + с cos((ny) +…

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Q: Assuming 9x° – 5y - 4xy = 0 defines y as a differentiable function of x, use the theorem dx - to…

A: Given: 9x3-5y2-4xy=0 We have to calculate dydx(1,1) by the given formula.

Q: -8. AB is tangent to to OC at point B. E 6. MZCDA - 8. MZDBA = 32

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Q: The value of 'b' for which u(x, y) = ebx cos 5y is harmonic

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Q: . Let w- e*² <in Cyz)- e Yz cor Cxz) Find - W X

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Q: A particle moves in the xy-plane so that its position at any time t is given by x= cos(5t) and y =…

A: According to the questionx=cos5ty=t3

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A: According to our guidelines, In case of multiple questions we are supposed to answer the first…

Q: Let f(x, y) = 2e sin(4) of д х of dy Check Answer

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Q: Given f(x,y,z) = cos(3xy) + sin(3z). Find fy(5,5,0). Leave your answer correct to two decimal %3D…

A: Partial derivatives

Q: 2, If fCx)=xInCx²+), find the ea uation for t4e tan9ent 1ine to fCx) at x= 3.Hint: Remember the…

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Q: 4. If u(x, y) is harmonic in some domain then find its harmonic conjugate v(x, y): (a) u(x,y) = 2x(1…

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Q: If f (x, y) = 4e4*cos (y - 1) , then the linear approximation at (0, 1) is %3D (A) 4+ 16x (B) 4+4x…

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Q: • Find a harmonic conjugate of u(r, y) = x² – 2x – y². Show that u(r, y) = e" cos y is harmonic.

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Q: (a) Suppose u(x, y) = 2x – x + 3xy? is harmonic, then find its conjugate harmonic v(x, y). -

A: The harmonic conjugate to a given function  is a function  such that is complex differentiable…

Q: 14. Evaluate 4y dx +3ax dy, where C is the trapezoid with vertices (0,0), (2,0), (6,2) and (0,2).

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Q: Prove that (y – æ)(y + 3x)³ = cx° is a solution to 3x2 – 2xy + 3y² )dx = 4xydy.

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Q: 10. If e is the acute angle between the surfaces xyz = 3x + and 3x - y + 2z = 1 at the point (1 -2…

A: We have to solve given problem:

Q: E). f(x, y) = sen?(r2 – 0²) + In(r0 + 0), r = Vx2 + y² 0 = arctan con -

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Q: (4) If z = f(r, y) where, a = e" cos v and y = e" sin v, show that (i) dz az e2u. (ii) az 2 2 dz -2u…

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Q: (a) Suppose u(x, y) = y³ – 3.x²y is harmonic, then find its conjugate harmonic v(x, y). %3D

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.