|1) Use a sine expansion to compute the solution of ( ди д'и at ôx? u(0,t)=u(1,t) = 0,t > 0, u(x,0) = x,0

Q: 4. Find a solution of the Bessel's equation of order zero.

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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: 8. Find the general indefinite integral (x³ -2² + x)dr; b) / (2" +-)dr, x > 0; c) (sin(3t) +…

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Q: 12. The integral of dy S Sec-' (5 ) + C 3 O ţSec-"' ( ) + C 3 Sin-l (플) G) +c 3 o Sim' (등) +C O…

A: This is based on simple formula of integration

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: 2. Use the Definition of Residue to show that: dz -ni 3 Jizl=1z² sinh z Where 0 < |z| < T.

A: Image is attached with detailed solution.

Q: Evaluate the integral. S 20x cos x dx Select one: O A.80 sin. 1 40x cos + C O B.20 sin n1)x+ 40x cos…

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Q: 4. (a) (b) Approximate the integral f x² cos xdx with n = 6 using the composite trapezoid rule;…

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Q: 1 Solve the integral equation S f(x – t)e-at dt - f(x) using Fourier transform x² + b² -00 to find…

A: Definition of Fourier Transform : Let f(t) be defined for all t∈(-∞,∞).Then the fourier…

Q: a)Evaluate y at position x=0.5  b)Evaluate the derivative dy/dx at position x=0.5. c)Using your…

A: For a given function of a single variable, its derivative represents at which rate the function is…

Q: Find the indefinite integral. |cos 2x ein 2* de sin 2x A) 2 +C ein 2x +C B) C) sin 2x e cos2x +C D)…

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Q: 2. The graph of y = cosh a is (a) concave up on its entire domain because y" = coshr > 0 for all r €…

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Q: 3. Find the linear approximation to g(z) = cos(z + 1) at z = 0. Use the linear approximation to…

A: The formula to find the linear approximation L(z) of a function fz at point z=z0 is: Lz=fz0+f'z0z-z0…

Q: Solve the problem. Integrate f(x, y) = sin(x2 + y2) over the region 0<x2 +y2< 100. O T(1 - cos 100)…

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Q: a) z = 4x²y-sen(3x - 4y)

A: # As per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: Which of the following is a left Riemann sum approximation of (x²) dx with n subintervals of equal…

A: Definition used-                  Left-hand Riemann sum -                   ∫abf(x)dx=…

Q: 5. Determine a function v(r, 0, ø) with Av = 0 _in B2 \ B1 satisfying the boundary conditions v(1,0,…

A: Use the cylindrical coordinates to determine the value of r. Now use the boundary conditions to…

Q: If In y = 3 sinh x show that dy (a) (1+r2)()² = 9y², dx (b) (1+ x²)- dy da? dy +x- – 9y = 0. da…

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Q: x³ + 6x2 + x+2 2 Tan Ve fx-1 ax + In{(x+ 1)*(x² – x + 1)"| + The resulting integral of -X+ -1 + C,…

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Q: 3- Transform the following differential equation xy"(x) + 2y'(x) + x y(x) = 0 u(x) to Bessel's…

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Q: 2. Find an asymptotic expansion of the solution y(x) to leading order that is valid on [0, 1] for…

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Q: Let f(x, y, z) = ln(x² + z) + y²e"z – cos(2yz). Find fzya- CoS %3D

A: use partial differentiation for finding the derivative.

Q: Use Laplace transform to solve the partial differential equation: 8² y dy dx² at +xt where y(x,0) =…

A: Solution

Q: In the integration of (cos y)^2 dy/(1 - sin y), what is the first step to solve the integral? a.…

A: Consider the integral: ∫cosy21-sinydy

Q: -) A) Find by using chain rule if f = ( , u = -1,x=-1 dx u+1 B) Evaluate the following integral:…

A: This question is based on chain rule differentiation and indefinite integral.

Q: If ulr. 0) satisfies V'u = 0 inr<1, u(1, 0) = S(e). show that Poisson's integral formula in R' takes…

A: Given, ur,θ=0 satisfies ∇2u=0 in r&lt;1. Also u(1,θ)=fθ To show ur,θ=1-r22π∫-ππfϕ1-2rcos(θ-ϕ)+r2dϕ…

Q: 12.)Express lim E(3(x;,)² – 5x, + 2)Ax as a definite integral on the interval [0,1]. i=1

A: The given expression is: limn→∞ ∑i=1n 3xi2-5xi+2∆x    on interval [0, 1]   The definite integral  on…

Q: 3. Find the residues of the functions 1- cosz а.) b.) c.) z'e at z=0; .5 e -1 and express the types…

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Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

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Q: B. Switch the order of integration: cos (x2) dædy. Just SET UP with the order switched. DO NOT…

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Q: The temperature u(x, t) in a rod of length - is determined from the following boundary value…

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Q: 7. Find the indefinite integral. (a) + sec , | dx = 4 In |x| + tan(x) +C (b) / )Lanr dx = In |In x|…

A: Given that: (a)∫4x+sec2xdx(b)∫1xlnxdx(c)∫x4-3x2+5x4dx

Q: | 45. Apply integration to the expansion 00 =E(-1)"x" = 1- x+x – x +.. 1+x n=0 to prove that for -1…

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Q: 1. Find an antiderivative of f(x) = 16–x². Then, use The Fundamental Theorem of Calculus %3D 4 to…

A: Given : fx = 16-x2

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: 6.3.18. Evaluate the integral. using U substitution integrate (cos^2(x)sin^2(x))dx

A: Hello, there are two functions present in the question. As the first function comes along with the…

Q: 2. Consider a homogeneous differential equation, p(т, у) dx + q(z, у) dy %3D 0, where the…

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Q: I need help with number 19 using the substitution formula. Can you please explain how it works…

A: We need to apply the method of substitution to compute the given integral

Q: 39. (a) Show that 1 1 > 0 2x V1+x² for x > 1. (b) Use your result in (a) to show that 1 dx 1+x is…

A: The compleet solutions are geven below

Q: 8) Evaluate the integral: f[(2 t² −5)i +cos j+(2+4)3 k]dt

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Q: A uniformly stretched string and fastened to two points 10 meters apart. The displacement of the…

A: This question is about application of Euler's equation

Q: 2. Use the Definition of Residue to show that: dz -ni 3 Izl=1z2 sinh z Where 0 < |z| < T.

A: Use the following formulae, to evaluate the given complex integral. Let z0 be a pole of order n,…

Q: Consider the IVP S2(t) = -1+cos(t + x(t)) cos(t + x(t)) x(0) = 0. Show that the ODE is exact and…

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Q: 3- Transform the following differential equation xy" (x) + 2y'(x) +x y(x) = 0 u(x) to Bessel's…

A: To Determine :-  Transform the Differential equation  x y''  x  + 2 y'  x  + x y x  = 0  to Bessel's…

Q: Compute the general antiderivative S a"-1 cos(x")dx COS where n > 1 is an integer greater than 1.

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Q: A:= {(x, y) € R² :1< log(x) < 2}. = {(x, y, z) E R³: sin(xy² + «/2) · cos(z) = 1} . (a) Decide…

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Q: 2.4- Find the Integration of SFex) -dx by Simpson's rule knowing that: i-6 > Use 1.5 3.0 72.0 1.0…

A: Simpsons' rule is given by- I=h3y0+yn+2y0+y2+....+4y1+y3+..... Where h is the step size, h=b-an…

Q: 2. Use the Definition of Residue to show that: dz -ni 3 Izl=1z sinh z Where 0 < |z| < T.

A: Cauchy residue theorem is one of the method to evaluate the integral of analytic functions.…

Q: [6 Let f(x) = |r| be approximated by sN(x) = }a0+E (an COS (") + b, sin (7). %3D n=1 Determine the…

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Q: 8. Find a recursion formula for the family of integrals I (n) = r I(1 x" sin x dx that starts with…

A: Given a family of integrals, I(n)=∫xnsinx dx