2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

Q: 4. Show that z = a' Zax if z = e-al cos(x-at) %3D

A:

Q: 1. A natural cubic spline S.on [0,2] is defined by (S.(x) S(x) = {x)=1+ 2x-x s,(x) = a+ b(x- 1) +…

A: The cubic spline is defined as: Sx=S0x=1+2x-x3 if 0≤x≤1S1x=a+bx-1+cx-12+dx-13 if 1≤x≤2 From…

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

A:

Q: Show that for some number a > 1, we have 1- <= In(a) <= a – 1. Hint: In(a) = S"dx, please check the…

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Q: 1. Show that c(t) = (e¬t cos t, e-t sin t), 0 <t <∞ has finite length and calculate its valu

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Q: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish…

A:

Q: (a) Suppose X U (0,n), where n is a positive integer X-[X] U (0, 1). (b) Use the result of (a) to…

A: De Moivre's theorem gives an axiom for computing the powers of complex numbers. We first gain some…

Q: H) 9f zyzec then prove thal nl x=4=Z, (alogen) %3D

A:

Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

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Q: Prove that v = In (x- 2)^2+(y-N)^2 is harmonic in every region does not include the point (2,N) and…

A: Given v=lnx-22+y-N2 where N=44. So, v=lnx-22+y-442                                    ........1 To…

Q: Find yas a fon chenof x if t. %3D with

A: This is cauchy euler equation and we will solve it by changing the dependent variable

Q: the Following is divergent

A:

Q: Find the in tervals on which f are: (il In creassing and decreasing Con cave uP and con cave down…

A: Here, the given function is: y=sin22x, [0,π]

Q: 2. Show that if zo is an isolated singularity of f(z) that is not remov- able, then zo is an…

A: We have to show if z0 is an isolated singularity of f(z) that is not removable, then z0 is an…

Q: Let C be the shortest path that connects z to mi and /= . (sinhz + 1)dz. Then |=-1-cosh t+ n(1-1)…

A:

Q: 2 Find the n th Lerm of the following Seq Uin Ln 2 2. Ln3 3. n= ,2,3,Y; -3 n=1,2,3,4 (2 Test the Con…

A:

Q: Let SINGLEDFA = { | M is a DFA and for all s e L(M),|s| = 1}. Show that SINGLEDFA is decidable.

A: Detailed explanation mentioned below.

Q: 6.1 let T be a l'iner transformatio from Mga into M,a Sub that %3D

A:

Q: 1+ By using the definition In(1+x) = / 1 -dt, show that for a certain range of values of r, t…

A: Solution:

Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

Q: 1. Use a Chain rule to find aw and ar aw , if w = x², x = sin(r²s) əs %3D

A: Given w = x2x = sin(r2s)

Q: 2. Show that g(x)= *** has a unique fixed point in [0,1] and find an approximation to the point that…

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Q: To prove that dx In /1+ (In x)² + In x + C x/1+ (In x)2 we use the substitutions

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Q: The Wronskian 1. Show that the functions e", cos x and sin x are linearly independent. 2. Show that…

A: 1. The functions ex, cos x and sin x are linearly independent.2. The functions 1, sin2x and cos2x…

Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

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. Evaluate the limits. e' cos (t – 4) dt - 4 (а) lim x2 x2 – 4 e2 + 2 (b) lim x00 e2x + 100

A: Solve the following

Q: 0, 2 is defined by A natural cubic spline S on 5. So()12x - 0 if 1 (1) S(r)2b(x1)c(x - 1)2d(x - 1)…

A: 5.Note that, the natural cubic spline must satisfy the conditions

Q: (2) Show that if dim(V) = k (so in particular, T has k distinct cigenvalues), then there exists v e…

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Q: Let C be the shortest path that connects a to ni and I = . (sinhz + 1)dz %3D O I-sinh rt+ T(i-1) O…

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Q: - Prove thas the scries 2+x2 3+X2 4+x niformcy conuergen2 in any interval?

A:

Q: If f(z,y, z) = I COs(2y – 2). compute and af and Te Əz =I Cos(2y -2). compute O = cos(2y2) I sin(2y…

A: If fx,y,z =xcos2y-z....icompute dfdx and dfdz

Q: Suppose that w = x² + y² – z² | And x = psin(4)cos(0), y = psin(4)sin(0), z = pcos(4) Use…

A: Simplify w by substituting values of x , y and z and then solve the partial derivative.

Q: Prove that v = In [(x-2)^2 + (y - N)^2] is harmonic in every region does not include .the point ( 2,…

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Q: Integrate the fuackion f=g regian bonndel by y the en6* y-o ver

A: The function is fx=xyThe region is bounded by y=x, y=6-x, y=0

Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

Q: Which of the following is a nonlinear second order nonhomogenous PDE a. xux-Uxx=tuxt O b.…

A: To find - Which of the following is a nonlinear second order non-homogenous PDE x ux - uxx = t uxt…

Q: Let f(x, y, z) = In(x² + z) + y²e#². Find fzyæ-

A:

Q: - Pcosø prove that dw xy is Cont s Lony) - the fun ction (-1, 2) = x*y+z and x = PSinf cse y=pSi- -…

A:

Q: 4. Suppose that u(x, t) is defined by a PDE and by initial values (at t = 0): du + to,u= u for all…

A: Suppose that u(x,t) is defined by a PDE and by initial values (at t=0)∂tu+t∂xu=u for all x,and…

Q: |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)…

A: According to the given information, it is required to compute the solution of the given equations.

Q: 15.If the Wronskian of f and g is t cos t − sin t, and if u = f + 3g, v = f − g, find the Wronskian…

A: Consider the provided information, Wf,g=fg'-gf'⇒tcost-sint ...... (1) Find the Wronskian of u and v.…

Q: s defined over ai at one point in tl асcordi

A:

Q: 3. C-1) " Conditionally absolutely codvergent divergent. Con vergent, 2n+ Sinin) or

A: Solve the following

Q: 3. Let V = eP cos(qr), with p = x In(xyz), q = z cos(Ty) and r = 2 tan-(y). %3D By using the chain…

A: Solve the given problem.

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: 1- Find dwi dt at the given value for w ==+2,x = cos (1), y = sin (1),z = w =-+ at = 3

A:

Q: if :z = cos 0+isin 0 2z² z²+1 prove that: 1+itan 0

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.