Q2) A) Verify that wry wyx for w x sin y +y sin x + xy. %3D B) Show that f(x, y) = e-2y cos 2x satisfies a Laplace Equation (+ = 0).

Q: Show that f(x, y) = e«" (A cos(ay) + B sin(ay)) satisfies Laplace's equation: = 0.

A:

Q: Suppose that r(t) = cos³ t i+ sin³ tj+et k, show that d dr (r.- dt dt d?r X at2 dr d'r. r.at at3…

A: Given: rt=cos3t i+sin3t j+e-t k we have to show that ddtr·drdt×d2rdt2=r·drdt×d3rdt3

Q: Find Wr 0 if W = f(x,y) x=r cos 0 and y = r sin 0, then prove that r Wr 0 =yfx + xfy + xy(fyy-fxx) +…

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Q: Solve using Laplace Transforms y"+4y = t-2 sin t; y(0)=-1, y'(0)= 2,

A: given differential equation isy"+4y=t-2sinty(0)=-1y'(0)=2applying laplace transformLy"+4y=Lt-2 sin…

Q: Show that z=tan-1(2xy/x^2-y^2) satisfies the Laplace equation

A: The given equation is, z=tan-1(2xyx2-y2). Obtain the partial derivative of z with respect to x as…

Q: Let z = e* sin(y+a), show that %3D ôx? oy?

A: Given, z=e-xsiny+a, then we have to show that ∂2z∂x2+∂2z∂y2=0.

Q: 1. Verify that the Laplace's equation: (i) u =x+3xy² (ii) u = sinxcoshy+cos.xsinh y f the uove…

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Q: þ/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+3y² Q/ find by using chain rule…

A: Since you have posted multiple questions ,we solved the one question for you  To get the solution of…

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: 10. Does the function f (x, y, z) = (x² + y2 + z?)-1/2 satisfy Laplace equation?

A: The objective is to check the given function satisfies Laplace equation or not.

Q: Let w(z, y, 2) = x² + y² + z² where z = sin(7t) , y = cos(7t), z = e'. dz dy dw by first finding dt…

A:

Q: Q2) A) Verify that wxy wyx for w x sin y+y sin x +xy. %3D R) Show that f(x v) = e-2y cos 2r…

A: See the details solution in below

Q: Show that the function z = arctan y/x satisfies Laplace’s equation                           ∂2z/…

A: Consider the following equation:

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: Q2) A) Verify that Wxy Wyx for w = x sin y+ y sin x+ xy. B) Show that f(x, y) = e-2y cos 2x…

A:

Q: f(x, y) =(e-2y)Cos2x

A: The Laplace equation is: ∂2f∂x2+∂2f∂y2=0

Q: Solve the Laplace equation subject to the boundary conditions, u,(0, y)= 0, u(2, y) = 0, u, (x, 0) =…

A: The Laplace equation is: ∂2u∂x2+∂2u∂y2=0 The given boundary conditions are: ux0,y=0, ux2,y=0,…

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

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Q: Solve by Laplace transforms. y – 3 | y(T)dr = 4e' sin(2t). %3D 0,

A: by Applying Laplace transformation and inverse Laplace transformation we get the answer

Q: Find Laplace treins form for Ý +4Y+3Y = e y(o) =y co) = 1 %3D %3D

A: Formula Used:L{y''}=s2Y(s)-sy(0)-y'(0)L{y'}=sY(s)-y(0)L{eat}=1s-aL{tneat}=n!(s-a)n+1

Q: Show that the function f(x, y, z) = x² + y? – 2z² satisfies the Laplace equation.

A:

Q: 10. Show that the function f(x, y, z) = e3z+4y cos 5z satisfies the Laplace equaton. Good Luck.

A:

Q: Find dw/dt if w = xy + z, x = cos t, y = sin t, z = t.

A: Given: w = xy+z,  x = cost,  y = sint,   z = t To find: dwdt

Q: Obtain the integrating factor of the non-exact DE, (2 + ycosx)dx + xcosxdy = 0. a. secx b. cosx c…

A: Integrating factor

Q: If z = x? + y?; x = t sin t , y = 2t cos t dz then dt at t = is

A:

Q: = prove that the following examples are satisfy Laplace equation , y) =(e-2y)Cos2x 2-f(x, y) = Ln…

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Q: 3) Find if w = 5 cos(xy) – sin(xy), x = ,y = t,z = t³ %3D dt

A: Let's find.

Q: Show that if the function f(x, y) = ecos-e-cosx, satisfies Laplace Equation.

A:

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

A: Given:           x3−2y2+xy=0,     (1,1)

Q: (c) Show that the function z = e* sin y + ev cos x satisfies Laplace equation ax? a2z = 0.

A:

Q: Utt || for 0 < x < T u(x, 0) = x? u:(x, 0) = }cos(8x) Uz(0, t) = u„(r, t) = 0

A:

Q: dw Let w =xycosz, where x = t°, y =ť, and z = arccost. Find dt

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Q: Show that the function f(x, y) = In /x² + y² satisfies the Laplace equation. %3D

A:

Q: Compute cos(x) cos(y) dx – sin(x) sin(y) dy, where c(t) = (9t, t2), 0 sts 1. (Round your answer to…

A:

Q: Q2) A) Verify that wry Wyx for w x sin y +y sin x + xy. s) %3D B) Show that f(x, y) = e-y cos 2x…

A: We have to verify that the equations satisfy the given conditions.

Q: Al/ compute 6(4+ ec©*) dx + (siny + 3x?)dy cosx Where c is closed bounded region lying in the first…

A: Given (A) The given integral is ∮4+ecosxdx+siny+3x2dy and the region lies in between x2+y2=4 and…

Q: b/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+ 3y² %3D

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: Show that f(x, y) = ea# (A cos(ay) + B sin(ay)) satisfies Laplace's equation: + - 0. dy?

A: Given,            f(x,y) = eαx(Acos(αy) + Bsin(αy))             Then show that  ∂2f∂x2+∂2f∂y2=0

Q: Consider z = x² + 3y², where x = sin t and y = cos t. Calculate dz / dt.

A:

Q: Verify that the function f(r, y) = In (x² + y²) is a solution of the Laplace equation %3D Zzz+żyy =…

A:

Q: Q2) A) Verify that wxy = Wyx for w = x sin y + y sin x+ xy. B) Show that f(x, y) = e-y cos 2x…

A: Here we have to find the partial derivative of the functions with respect to x and y seperately…

Q: (c) Show that the function z = e* sin y + e cos x satisfies Laplace equation azz azz = 0. ax? ay?

A:

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: Show that the function u(x, y, z) = e3x+4ysin(5z) satisfies the Laplace equation in R3

A: A function u(x,y,z) is said to satisfy the Laplace equation in ℝ3 if u(x,y,z) satisfies…

Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

A: Given that dydt=yβ-αy and y=y0 at t=0 We have to find the solution of the above equation.

Q: Verify the Pierre-Simon Laplace equation ∂2f/∂x2+∂2f/∂y2=0 with f(x,y)=ln∣x2+y2∣

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Q: find the general solution of the dipperential ation given below. y+ 2xy -xe-r?

A: