Show that the function f:R –R givenby f(x) = log, (x+vx* +1),a>o,a±1. is invertible and find its inverse.

Q: find the inverse Laplace transform of the given function by using the convolution theorem.…

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Q: Express f(a) by the Fourier series f(z) = 2, ーTくェS0 0< ST' where f (x) = f(x+27). %3D

A: Hi! As per norms, we will be answering only the first question. If you need an answer to others then…

Q: Find the integration norm of the given polynomial on C(0, 1]. f(x) = -6x – 4 || f|| = Ex: 1.23

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Q: Determine weather the statement is true of false. Explain your answer.

A: Consider the given differential equation dydx=1x2+1 Use separation of the variable method to find…

Q: Show complete solution. Expand, that is, perform the indicated operations on the differential…

A: We have to find out the operation

Q: Find the formula for the function represented by the integral 1 -dt= 5t+1

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Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: By the odd extension method applied to the function f(x)=e-2=, where r > 0, the Fourier integral of…

A: Given function fx=e−2x where, x&gt;0 We have to find the Fourier integral by the odd extension…

Q: By the odd extension method applied to the function f (x) = e¯2x, wherex > 0, the Fourier integral…

A: Given the function  fx=e-2x

Q: (a) Find if y = y (ln r). (b) The function f(z) (In r) is an invertible function (you do not need to…

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Q: Show that 1, x, and x2 - ½ are orthogonal, when the integration is from x = -1 to x = 1. Write f (…

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Q: Consider f(x) = x^7 + 2x^4-1, 0 = 1. Integrate f over the given interval of x, using Newton Closed…

A: Given, f(x)=x7+2x4-1,  x∈0,1 if we let h=0.25 from newton-cotes formula ∫abf(x)dx=h2f(x0)+f(x1)…

Q: Consider a continuous-time signal g(t) with its Fourier Transform as below: (2 sin(w) G(jw) =…

A: As per the question we have to first find the expression for the signal x(t), and then we have to…

Q: Find the inverse Laplace transform of the given function using the Convolution theorem. 1 (5) F(s)=…

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

Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

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Q: 89. Use the chain rule to prove the formula for integration by substitution: |f (u(x))u'(x) dx =…

A: To prove ∫f'ux u'xdx=fux+C by substitution. Solution: Let substitute t=ux. Differentiating both…

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ € R² f2(x, y) = x'y, x, y E R" f3(x) =…

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Q: Evaluate the integral f e² In (x²+1)dx

A: ∫e2ln(x2+1)dx

Q: f(t) = {b 0<t<3 otherwise

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Q: Find the Laplace Transform of f f(x) = 1 t Osx<1 x 2 1

A: We will find Laplace transform of given function.

Q: Let f: (−π / 2, π / 2) → R be given by f (t) = tan − 1 (t). (a) What is Taylor's polynomial P1 (t)…

A: Consider the given function. Find the derivative to determine the first order of the Taylor's…

Q: Use the Beta and gamma function -1 n! 1- Prove that x™(ln x)"dx (m+1)n+1

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Q: Find the weighted integration norm (f, f) = | f(x)f(x)w(x)dx of the given polynomial on C[0, 1].…

A: Norm of a vector is defined as f=f,f.

Q: Apply (3) to establish the formula for the Laplace transform of an integral, F(p) L|f(x)dx 1. and…

A: Given the equation, ℒy'=pℒy-y(0)  ⋯⋯(1) Let ℒf(x)=F(p) Also, let yx=∫0xfxdx Then, from equation (1),…

Q: Use the Laplace transform to solve the given integral equation. f(t) + f(t) dt = 2 f(t)

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Q: HW.8. Use farseval's theorem to find if f(t)=A e-alti Ans: A?

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Q: Find the Fourier cosine integral representation of the following: f(t) = {to 0 < t < 1 elsewhere

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Q: f(t) + f(t) dt = -2 f(t)

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Q: Determine if the following functions are linearly dependent or independent by determining their…

A: The Wronskian of a set of function y1, y2, y3 is defined as, If the Wronskian W is non-zero then…

Q: Find the Fourier integral of the function f(x)= e* when x> 0 and f(-x) = f(x) for k > 0 cosix and…

A: Fourier integral, for even function

Q: 2n Jnsi (x) = " J,(x) – J(4)

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Q: Consider the function Alt) = -2t + 10. Calculate At)dt by using 5.1 a graph and areas 5.2 a definite…

A: From the given function: ft=-2t+10 As we know that: The standard equation of the line…

Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D

A: As it's mentioned in the question to answer between a) or b), so showing steps for only one part. b)…

Q: The expanded form of the differential operator D(xD -1) is A xD +1 O xD? -D © xD? -1 (D) O xD?

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Q: determine the inverse Laplace transform of the given function.

A: To determine the inverse Laplace transform of :…

Q: Obtain the Fourier transform of the double pulse defined by f(t) ={ 1 ,1< or equal to x < or equal…

A: Fourier transform: The Fourier transform of function f is given by, F(ω) = ∫-∞∞f(t) e-ωjtdt. Given:…

Q: Prove that the Generating function of Bessel function i.e. £1J„(x)t" = e*/2(-1/2), %3D

A: Let ϕx,t=ex2t-1t. Let us prove that ϕx,t is the generating function of the Bessel function…

Q: Find the Fourier transform F(@) = | f(t)exp(iwt)dt of the function 0, t 0. You might want to use…

A: The given function is ft=0t&lt;01βtt≥0, where β&gt;0. We know that the Fourier transform of a…

Q: Consider the function f defined on [0, 1] such that / (0) = f(1) = 1.f(1/5) = {(3/5) - 2, and /(2/5)…

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Q: Determine the expression of the integrating function if it is known that: e) S; f(x)dx : - sen(9) +…

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Q: Show that the following functions all grow at the same rate using properties as x co: fi(x) = x°,…

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Q: Assume x and y are functions of t. Evaluate dt for 4xeY12-In 729+61n x, with the conditions -6, x-3,…

A: Given that 4xey=12-ln729+6 lnx and the conditions are dxdt=6 , x=3, y=0  find dydt

Q: Find the weighted integration norm (f, f) = | f(x)f(x)w(x)dx of the given polynomial on C[0, 1].…

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Q: find the inverse Laplace transform of the given function by using the convolution theorem.…

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Q: Consider the following. dt (4 + t? Let t = 2 tan(0) and determine the following. 2 sec-(0) dt = de t…

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Q: Given F(x,y) = &lt; 3x^2e^y, x^3e^y + 1/y &gt; is conservitive, Find the potential function phi(x,y)…

A: Given: F(x,y) = &lt; 3x^2e^y, x^3e^y + 1/y &gt; is conservative.

Q: By the odd extension method applied to the function f(x) = e-2=, where a > 0, the Fourier integral…

A: The given function is, f(x)=e-2x,x&gt;0

Q: verify that the given function is in the kernel of L. Q. y(x)=sinx +cosx, L = D3+D2+D+1.

A: The expression is yx=sinx+cosx, L=D3+D2+D+1.

Q: Evaluate using Laplace Transform, find a: sin(t + a) cos(t + a) f(t) = |. 3 dt 8 %3D %3D e2t

A: Given: f(t)=∫0∞sin(t+a)cos(t+a)e2tdt=38 To find: Find the value of a by find integration using…