Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set in motion from its straight equilibrium position with Constant velocity vo at each point is 2} sin{ U (x, t) -sin m=1 (2n-1)² 1 (2n-1) (2n-1)пс C

Q: Evaluate the line integral / (9x + 10y)dx + (5x – 6y)dy along the curve C : a = 6 cos t, y = 12 sint…

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Q: Let y = 2x + 2x +3. If Ar = 0.2 at a = 2, use linear approximation to estimate Ay %3D Ay Submit…

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Q: Use the total differential dz to approximate the change in z= 6-x² – y² as (x, y)moves from the…

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Q: 4. Let f(x, y) = 2TY x = In 2 g and y = In s. . a. Find the equation of the tangent plane to f(x, y)…

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Q: then show that u and v satisfy the Cauchy-Riemann equation everywhere for ( i) and for all z # 0 in…

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Q: 2. When object P, with mass m, , moves at v, and object Q, with mass m,, moves at v,, the velocity…

A: Given,For P, mass = m1  and velocity = v1Let, velocity after collision is u1For Q,mass = m2 and…

Q: Find the maximum and minimum values attained by f (x,y, z) = xyz on the unit ball ² + y² + z² < 1.

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Q: 8. Find the flux of F = (2x – 5y², 3y, x) out of the sphcre of radius 2 centered at the origin.

A: The flux of the vector space is given by the formula, ∫S∇.F dS where S is the sphere of radius 2…

Q: 10. What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F =…

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Q: Glven: XY and ZW Intersect at polnt A. Which conJecture Is always true about the glven statement?

A: as given that XY ↔ and ZW↔ intersect at point A. so, here we understand few terms which is given in…

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A: .

Q: Evaluate the line integral / (7x + 3y)dx + (2x – 8y)dy along the curve C: x = 8 cos t, y = 16 sin t…

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Q: Find Fxx and Fxyz of: XX хуz y2+Cos (xz) f (x, y, z) : 2 Siny

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Q: Find the differential dy: y = In(sec² ( 3ª²–20+5 312-2x+5 O (E) None of the choices dy = 2 (x – 1)…

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Q: Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y =…

A: Given: w=x2e2ycos3z x=cost y=lnt+2 Putting x, y and z in w, we get…

Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

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Q: Calculate the work done by a constant force field F on a particle that moves once uniformly along…

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Q: Resuelve la ecuación integro-diferencial y'(t) = cos t + y(7) cos(t - 7) dT, y(0) = 1 %3D

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Q: Let w = x^2e^2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y…

A: Given  By chain rule of differentiation , we have

Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: A homogeneous string is stretched and its ends are at x = 0 and x 1. Motion is started by displacing…

A: The boundary conditions are y(0,t) = y(l,t) = 0 and initial conditions are y(x,0) = μ0sinπxl and…

Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Find the orthogonal trajectories of r= a sin? 0 where a be the arbitrary constant.

A: We have, r=a sin2θ .....................................................(1) Differentiating w.r.t.…

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Q: Show that f(x, y) = ln(x2 + y2) solves Laplace's equation, ∂2z ∂x2  +  ∂2z ∂y2  =…

A: Given Laplace’s equation is

Q: Let x,(t)-v(t)+sin(r), x,-vt) then x,(t)= xN-cosit) Select ones True, False

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Q: Suppose that z = xy² and x and y are differentiable func- tions of t with x =2,y = when t = -1. Then…

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Q: 22. u- Vr? + s, r- y + x cos 1, s -x+ y sin t; ди ди ди when x= 1, y = 2, 1 = 0 ar ay at

A: Given: u=r2+s2  r=y+x cos t and  s=x+y sin t By using Chain Rule we have to find: ∂u∂x, ∂u∂y ,and…

Q: sin(t) et, X2 sin(t)- cos (t) and X3 D sin(t) + cos (t)] 0. Given that X, cos(t) sin(t) + cos (t)|…

A: The problem is wrong, three vectors are not linearly independent.

Q: Find the potential function for F(x,y,z) = (x^-1z)i + (y^-1z)j + ln(xy)k. Show all steps and explain

A: Potential function of  F, is f such that ∇f=F.

Q: Consider h(x, Y, z) = cos (xy) + eY% + ln (xz). Determine the directional derivative of h at the…

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Q: Find the Second appoximation for the System. X Sin y at (xo,,)= (1, 1). y=Cosx

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Q: 2. Prove that the directional derivative of f(x, y, 2) = ry sin z at (1,2, ) in the direction v = i+…

A: Given:        fx,y,z=xy sinz         x,y,z=1,2,π2    direction v=i+2j+2k   Differentiate the…

Q: Let y = tan(3r + 4). Find the differential dy when z = 2 and dr 0.1 %3D Find the differential dy…

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Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: 1. Show that the problem may be formulated in terms of the velocity potential (r, 0) satisfying r dr…

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Q: Let z(x, y) = 2x2 3y2 where x = 2s – 4t & y = 9s + 8t. | dx dy ду az Calculate ds dz & and using the…

A: We have to find all first order partial derivatives.

Q: Find the divergence of F(x, y, z) =(a -1 Y, y? + 1' 3xz2 + 2y ) at (2,0, – x COS

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Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: 2. Find the complex potential function f(z) = u + iv, given that 2u +3v = e*[cos(y) - sin (y)]. Also…

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Q: Find div (e" grad u x 7) u=x-y ex at (2,1,0) where and =(x²+1)i + sec¯¹(x² −y)j + 3y²zk

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Q: 2 Find the orthogonal trajectories of r=asin? 0 where a be the arbitrary constant.

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Q: 10. Find the work done by the force F=vx +2y y - 7 for a particle that moves between points (-1,0,1)…

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Q: 3. A particle is located at a distance x meters from the origin, a force of x +2 newtoms is acting…

A: To find the work done

Q: Show that g(x) = cos 2 =x+3 ™K(x, t) g(t) dt 0 || sin x cos t, 0≤x≤t Y sin texr

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Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: the equation we have  (x3-y4)6+ln(x2+y)=1

Q: 4. f(z) = { |z|* z = 0 verify that the Cauchy-Riemann equations are satisfied at the origin z =(0,0)…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.