[#1] Exercise 7.46. Prove that if y = y' then y = ky' for some scalar k.

Q: -1 < x <0 The Fourier integral of f(x) = 1, 0<x<1 0, otherwise 00 (1 – cos(w)) sin(wx) dw lw 1. 00…

A: Given that fx=-1-1&lt;x&lt;010&lt;x&lt;10otherwise The Fourier integral of f(x) defined on -∞,∞ is…

Q: 10) Find the potential function for the vector field F=-i-4j - k. z² Then compute the path integral…

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Q: (c(2z+y), %3D ue of c in the joi F of W.

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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)…

A: Let f(x,y)= 2xy , x=t/ln2 and y= lns Here we have to find the equation of the tangent plane to…

Q: Integrate counterclockwise around C. (Show the det に+1 20. ‚ C: |z − 1| = 3 - ²-iz + 2

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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: 4x Find the linearization L(x,y) of the function f(x,y) = e cos (3y) at the points (0,0) and0,

A: Answer is given below

Q: 2. Determine whether the functions f1(x) = e2", f2(x) = e=2#, and f3(x) = sinh 2x are linearly…

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Q: 2. Let V be the set of all the solutions of the differential equation y" + w?y = 0 in the form y =…

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Q: Use the direction field given below to sketch the graphs of the solutions that satisfy the given…

A: To find- Use the direction field given below to sketch the graphs of the solutions that satisfy the…

Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

Q: 2x* + ax + b Part A: Consider the function f defined over]-0; 2[ U]2;+«[ as : f (x)= where a x- 2…

A: Given data: The given function is f(x)=(2x2+ax+b)/(x-2). (a) The expression for the derivative of…

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

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Q: Derive a Dirichlet principle for the elliptic operator L in divergence form with A e C² and b= 0.

A: Given elliptical operator L~ in divergence form with A∈C1 and b=0. Suppose A∈C1Ω of the second order…

Q: Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many…

A: Consider the given path rt = cost , cos2t , sint  Then, xt=cost , yt=cos2t , zt=sint

Q: Assume x and y are functions of t. Evaluate dt for 2xy - 2x + 4y3 = - 56, with the conditions = -…

A: Given      2xy-2x+4y3=-56 →           1      dxdt=-16, x=4, y=-2

Q: Express a general solution to 2x (1 — х)у" + (3 — 10х)у — бу %3D 0 Using Gaussian hypergeometric…

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Q: Find the orthogonal trajectories of the families of r e, where a being a parameter. (0)

A: Orthogonal trajectories are another family of of curve which are perpendicular to the the original…

Q: 4.5.1. Compute the best quadratic approximation to f(r,y) = e" cos y at the point (0,0), f(h, k) z…

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Q: Compute N(0) if r(t)= and prove that T(0) and N(0) are orthogonal

A: Given rt=&lt;t sin t+cos t,2t2,sin t -t cos y&gt; Then vector is given by r→t=t sin t+cos…

Q: 1. The position of a particle in space at time t is de 3 -k: =' - 1 1+t- j+ t+1 a) lim Okelocity…

A: The position vector of the particle is r(t)=et-1ti+1+t-1tj+3t+1k

Q: az az and as Use the chain rule to find at (=) Z = tan V u = 8s + 5t, v = 5s – 8t az as az at

A: Topic:- derivatives

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

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Q: (iv) Find the work done by the force field F(x, y) = x²i+ yeªj on a particle that moves along the…

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Q: 2. Find the linearization L(x, y, z) of f(x, y, z) = x² + xy + yz + 22 at the point (0, 1, 2).…

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Q: (b) Prove that sint -dt > 0 t+1 for all a > 0.

A: Given:- ∫0x sintt+1dt &gt;0

Q: 1.6 Evaluate the line integral f x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

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Q: 2. Use the Fourier transform to solve this transport equation for u = u(x, t), defined on the whole…

A: Given that                       2ut-3ux = 0 and  u (x,0) = e-x2            We can write this as…

Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Show that the function f(x + iy) = Vlæ|ly| satisfies the Cauchy-Riemann equations at the origin but…

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Q: Consider the vectors x(¹) (t) = ( ) and x(²) (t) = ( 6 ). 6t (a) Compute the Wronskian of x(¹) and…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Exercise 5 3x + ax + b x² +1 Conşider the function f that is defined over IR as: f(x) = Where a and…

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Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

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Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Find the linearization L(x,y) of the function f(x,y) = e* cos (9y) at the points (0,0) and ...

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Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

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

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Q: Consider F(x, y, z)=e-2 yi-y³/²zx²j+ln(22)k and calculate the divergence of F.

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

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: (z - 1)dt + (1 – t)dz %3D

A: (4) Given differential equation is z−1dt+1−tdz=0......1. We have to determine the whether the…

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

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Q: 2. Repeat the above question with a Hamiltonian of the form H(p, T) : +V(#), 2m where p(t) and (t)…

A: repeat the above question with a hamiltonian of thr form Hp→,x→=p→22m+Vx→ where p→t , x→t are vector…

Q: 2. Consider the PDE u -t ux+ tu =0_for - ∞ <x< ∞ and 0 <t< ∞ with u (0, x) = sine x. Determine u (t,…

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Q: 6x Find the linearization L(x,y) of the function f(x,y) = e cos (y) at the points (0,0) and

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Q: 1. Find the linearization of r – z at a = 2.

A: Find the linearization of x3-x at a=2.

Q: 2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1),…

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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.