(a) Show that {e¬*,xe¯} is a fundamental set of solutions of y" + 2y' + y = 0 on (0, 00).

Q: 6. (a). [3-2 စာ ၅၊ = Let A 1-1-2 = 0-2 Compute A-1 ***** (b) Use part a) to fin all solutions to the…

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Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: Find the coefficient of x10 in (1 +x+x²+x³ + •..)" Find the coefficient of x' in (x +x6+x² + • ·…

A: 1)Given expression1+x+x2+x3+....nThe series 1+x+x2+x3+.... follows a geometric progression with…

Q: (3) positive Verify that the fourth-order ode u - u" - u = -wu has a solution with w 3 sech( /20…

A: Given: u''''-u''-u3=-ωu  ⋯1 We have to verify this fourth-order ode. Also determine ω.

Q: Give algebraic proofs of properties (1)u + (-u) = 0 ,(2) (c + d)u = cu+ du

A: Let be a vector such that  Multiply  with -1. Adding the two results,  Thus,is the additive…

Q: A student is solving a Cauchy-Euler equation and writes the following auxiliary (characteristic)…

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Q: 4) Solve the Cauchy euler equation x- 3x2+ 6x2-by = 3 + Inx dr

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Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3) (2z…

A: 3. Given function is 2z+1zz3-5.

Q: 1 1 T+ 2 1 B T+ - T(2)"-1 3(T, T)

A: To prove  βT+12,T+12=πT24T-11βT,T It is known that  βm,n=ΓmΓnΓm+n Now, for m=n=T+12,…

Q: 6. Show that when ) < |z - 1| < 2, e - ( - 3) (2 - 1)(2 – 3) Complex Analysis n=0 (z-1)" 2+2 2(z-1)

A: We use Laurent series to prove given example.

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: Let p(x) =-2x* + c,x' + c,x + c,x + c, x ER be a polynomium whose graph insersects the points (-2,…

A: Given. We are given a fourth degree polynomial and also four points on the curve of the polynomial…

Q: Show that |1 - (x, + x,) - (x, + x,) X,*z = (x, – ,) (x, – y2) (x2- ,)(x2 - 2) 1 -(), + x2) 1 - (y,+…

A: See the attachment

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: 5. Let f(z) = 3z* + iz3 + z + 7. Prove that all roots of f (z) are in the annulus 1 < |z| < 2.

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Q: (x + 3)³ 12 16 11 f(x) Consider the function 4(х — 1)2 4 1 (x – 1)2 4. Note that - (x – 9)(x + 3)²…

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Q: Let C be the shortest path that connects t to ni and I = S. (coshz + 1)dz. Then I=0 1--sinh π+…

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Q: 6. Show that when 0 < |-1|< 2, (z-1)(z-3) Complex Analysis (z-1)" 2+2 =-36 z−3) ² A=0 2(z-1)

A: We use substitution and Laurent series expansion.

Q: Prove that H(x, y) = (x, –y) provides a conjugacy between x' = (4 })x and Y'= (; 7). X and Y' = 1 -1…

A: Given: X' =11-11XY' =1-111Y To prove: H(x,y) = (x , -y) provides conjugacy between both of the above…

Q: 3. A fundamental set of solutions of x' : 2 x is: 4 (a) {e (» {* (1)· [- (}) * (})]} () {**(;)· [-…

A: First we will find the eigenvalues and eigenvectors of the coefficient matrix of the system then…

Q: y"+y' + (a) -5 x-b = 0, Xo = 2 is 5. (b)-3 (c)-2 d)-8 Q7) L(y") = a) s²L(y(x)) + sy(0) - y'(0) c)…

A: We have given the answer in the next step. Hope you understand that

Q: 2) Consider the following system: - ax (1-)– Bxy (): x' - Вху K. y =ry (1-) µx. Find the equilibria…

A: 2) Consider the following system: x'=αx1-xk-βxyy'=γy1-yμx Find the equilibria and analyze their…

Q: (5) Compute -2 67 using Euler's criterion.

A: Objective: To compute -267. Concept used: Euler's criterion, is a result from number theory that…

Q: Let f(x, y, z) = x³y5 + z4 and x = (a) Calculate the primary derivatives of, of of дх' ду'dz' (Use…

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Q: show the function is continous dis continou or 2=(2x+3 3メ→1」Xハz F(x)ニ

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

Q: The function f(x)=/25-x' is contionous on

A: It is continuous function.

Q: In what form should the particular solu tion be sought for the equation y(1) + 2y" + y" + 2y' =…

A: Consider the equation y4+2y'''+y''+2y'=xe-2xThe characteristics equation of the homogenous equation…

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (2² + 3z – 5)/z3 (3) (2z…

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Q: Prove that H(x, y) = (x, –y) provides a conjugacy between 1 X' = 1 X and Y' Y. 1 1 1 1

A: We are Given that :  H(x,y)=(x,-y) X'=11-11X , Y'=1-111Y The Matrix H is corresponding to the…

Q: Consider the following bijections on R: f (x) = - g(x) = x – 7 Then (f o g)¯'(x) =

A: Given, fx=-x,gx=x-7 Find the value of f∘g-1x.

Q: If the invertible function f (x) satisfies f (12) = 6, then which of the following statements is…

A: Option C is correct option.

Q: Cauchy - Eulen : 7) x*y"-hs xy'-1sy=o

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Q: 4) fi(x) = cos 2x , f2(x) = 1, f3(x) = cos² x %3D %3D

A: