3. Solving the following problem using Fourier transformation method. du d²u a² -= 0, (t> 0, -∞0 < x <∞) Ət əx² u(x,0) = sin x -

Q: Find the constants a and b such that the complex function f(z) = x² + ay² - 2xy + bx²i − y²i + 2xyi,…

A: A function f(z)=u+iv is analytic if ∂u∂x =∂v∂y and ∂u∂y =-∂v∂xGiven…

Q: as Calculate (x sin y + e). ax²əy³

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Q: 1 (-0 + 7) - 1 Sketch a graph of the function f(0) = -3 sin(-0+

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Q: 7. Using Lyapunov functions, show that the origin is unstable for the ODE i = -x3 ÿ = -y3

A: The solution is given below

Q: 1 f(x) = 5 sin (-x - π)

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Q: Find the Fourier coefficients of the periodic function f(x)=x-2x², (-<x<n); f(x+2) = f(x)

A: We have to solve by Fourier series.

Q: 1. F{(t³ – 3e*)²} 2. Ąe-2"cos3t + 5e-2'sin3t} 3. A(t² – 4t + 2)e³} 4. {3t*e¬6*cost} 5. A4t²sint –…

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Q: Sketch a graph of x = -3 + cos(t), y =1+ sin(t), 0 < t < 27 COS 5+ 4-

A: Given: x=-3+cos(t)y=1+sin(t)

Q: * (). Partial Differentiation. If u = In(x² + y²) and z = sin-1 evaluate the following: a²u 1. əx2

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Q: (a) Show that the polynomial of degree 3 is differentiable in the complex plane, also find f'(z).…

A: For part (a): It is required to show that the polynomial of degree 3 is differentiable in the…

Q: Question 27 Determine the second derivative of 6 y - xy2= 1 A y " = 2y³ ( 6 – 2 x y ) -2 ( 2 – x y (…

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Q: lim tanx i. x→ 0 lim In(1+ x)] 11. x→ 0 lim sin ax i1. where a and b are constants. r0sinbxr

A: We are only allow to to do first question and three subpart of question So if you need rest of…

Q: Question 2 Find the linear approximation of the function at the given point. f(x, v) = 3 sinx - 7…

A: F(x,y)=3 sin x - 7 cos y at ( 0 , π/2)

Q: Problem 7/ Check whether the function u(x, y)=2y(x-1) is harmonic and obtain the conjugate function.

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Q: у" - 2у" - у +2у %3 0, У(0) %3D 1, /(0) — -9, у"(0) — 13. y(x) =

A: Given differential equation is y'''-2y''-y'+2y=0,  y(0)=1, y'(0)=-9, y''(0)=13

Q: Now, 2 x (+) dt 2 50, d2x(+) dt 2 -1 d2x(1) dt 2 is +2 J2x(+) ct 2 + 1 +2 = -15(++ 2) + 25 (1-0) +…

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Q: [Motion in R³] A drone has an initial position of (3,1, 2) and an initial velocity of (-2,5,1). The…

A: Here we can use the following formula to find displacement function: vt=∫at dtst=∫vt dt

Q: dz Use implicit differentiation to find dx In(x+z?)+yz+6x?=7 2(7x2+722+1) 2(3yz + 3x?yz + 2)…

A: We can solve these question using the properties of partial derivatives as below

Q: Find Solution of Homogeneous linear ODE s (D°, 60 +131 Jy

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Q: State the definitions of direct and inverse Fourier transforms. Then, compute the Fourier transforms…

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Q: 3. a) Find the Fourier transform of g(x) given that g(x) = 3e¬4|x+2| b) Using the formulas for…

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Q: H(s) = 1 s³ +bs² (use integral rule)

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Q: (ex, if 0 a 3 Represent the function f(x) = { as Fourier cosine integral.

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Q: [8] Find the complex Fourier integral representation of f(x) = e2x 0 |x| < 1, |x ≥ 1.

A: The given problem is to find the complex Fourier integral representation of the given function.

Q: find the derivatives complex funcation. {tanh¬'(iz +2)}-1,

A: fz=tanh-1iz+2-1=1tanh-1iz+2 It is known that tanh-1x=11-x2 so it follows:…

Q: 1. ('88 BC) The length of the curve y = x³ from x = 0 to x=2 is given by 2 2 A) [√₁ + x ²dx (B) √ √…

A: Given,.Formula for length of the curve is,

Q: ) If F(ƒ) = ƒ(§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is…

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Q: v" - 4v" +5v' - 2y = 7t-5. v(0) = 0. v'(0) = 0. v"(0) = -2

A: