6. (a) Let T: L²[0, 2π] → L²[0, 2π] be given by 2п Tf(x) = -Ĵa cos(xt) f(t) dt. Show that (i) T is self-adjoint. (ii) cos x and sin x are eigenvectors of T.

Q: Consider any(n) + an-1y(n-1) + + a1y' + aoy = f(x) where an, ...a1, ao E R, where an 7 0. ... (a)…

A: consider anyn+an-1yn-1+.....+a1y'+a0y=fx            ...i where an, ......., a1 , a0∈ℝ where an≠0 (a)…

Q: 3i +2V2 - 5n -4i +6(1 +a T/4 [(3sec2 t)i - (5 + sin t)j - (6sec t tan t)k)]dt Find 0. 2 2-57T-4 3i +…

A: ∫sex2xdx =tanx+c∫(constant)dx=x+c∫sint dt =-cost +c∫sect tant dt =sect+c tanπ4=1cosπ4=12secπ4=2

Q: cos(rw) 1+w²

A: Fourier Cosine Transform or Infinite Fourier Cosine Transform: Definition: Let be a function defined…

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A:

Q: If r(t) = = cos(-3t)i + sin(−3t)j – 10tk compute r' (t)= i+ i+ (cos (-3t))/3 and fr(t) dt= (-sin…

A: r(t)=cos(-3t) i+sin(-3t) k-10tk

Q: (c) Given T" = 6x²y³ – cosx + 5y² – siny + 10x² – e-3x," determine (1) T' = S(T")ax T' = S(T")ay…

A: GivenT"=6x2y2-cos x+5y2-sin y+10x2-e-3xwe know…

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A: Two distinct functions f and g are said to be orthogonal on [a,b] if &lt;f,g&gt;=0   , where &lt; ,…

Q: O [-5, -1) U (1,6] O (-5,6] O [-5, 6] O (-5, -1] U [1, 6] Find the domain of r(t) = (ln(t + 5),…

A:

Q: Determine the Fourier transform of f(t) = te-2t u(t), where

A: Given ft = te-2tut where ut =0     t&lt;01    t&gt;0 The Fourier transformation is given by Fw…

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A:

Q: Let f(2) := 25. Show that f is invertible in a neighbourhood of i and calculate (f-1Y(/(i)).

A: Let f(z) = z5, show that f is invertible in a neighbourhood of i and calculate(f-1)' f(i)

Q: Show that the integral equation TT g(x) = 1 (sin x sin 2t) g(t) dt has no eigenvalue

A: Solution:-

Q: 2) find the fourier transform x(t) = u(t) x(t) ulzt) x²(t) = u(+ + 5) -

A: xt=utxt=u2txt=ut+5xt=ute-jw0t

Q: Compute r"(t) and r'''(t) for the following function. r(t) = (7t2 + 4,t + 3,6)

A:

Q: 0, - 00<t<0 e at, 0<t<∞ Find the Fourier Transform of f(t)={ a is positive constant

A: The Fourier transform of the function f(t) is represented as F(k), it is defined as ∫-∞∞e-iktf(t)dt,…

Q: show that the function y = Ae- Is Eigenfunction for this operator d? 2 d + + dx2 'x dx 2a Where A…

A: The solution is given as

Q: 2+5( jw) (jw)' +3( jw)–10 a- F(w)=-

A:

Q: 5s Evaluate J|2 sin s Jdtds 0 0 I[ 2 OA. 3 (1 – sin(125)) O B. 3 (1+ sin(125)) (1+ cos(125)) 3 OC. 2…

A:

Q: a) Find the Fourier transform of f(t) = e-tu(t) + 4 t² +9.

A: As per the policy of bartleby the provisions is to solve only one question at a time

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A:

Q: Find the Fourier transform specified in part (a) and then use it to answer part (b). (a) Find the…

A: In part (a) we have to find the Fourier transformation, Given, f(γ,p,t)=e-γtsin pt  t&gt;0,0…

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

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A: